Statistics Lecture 5.3: A Study of Binomial Probability Distributions

in 5.3 we deal with something called the binomial probability distribution and that kinda music manager for you you see a lot of similarities to what I'm talking about and what we have over there or at least the idea of flipping that coin so the binomial I know let's buy me two no mule main two or outcome to outcome probability distribution I know distribution in this case by of course means to know meal means for us it means sums like name term outcome for us means outcome what this is is a probability distribution where there are only two outcomes two outcomes something considered a success and something considered a failure now you can make a lot of things into a binomial distribution member role in the diet and you have actually six outcomes there right but if you say thanks if you say my success is rolling a form everything else would be a failure doesn't that only have two outcomes you either get the four or you don't does that make sense to you so we can make things into binomial distributions even if they have more than that number of number outcomes we qualify them one would be a success and one would be a failure 200 if you would okay so that's that's what this means we categorize our two outcomes as a success or a failure that way there's only two of them by number so we type the probability distribution there where there are only two outcomes success and failure now there's a few things you need to know about a binomial probability distribution before we actually go and do any examples and that's some some vocabulary some symbol civilization notation and what it takes to even have this binomial distribution here so here's the rules for this thing first rule you have to have a fixed number of trials which means you can't go and do this procedure for eternity it's got to end somewhere for instance this is a this is you're gonna find out that this was a binomial distribution idea you're flipping a coin you looking for heads heads would be a success and tails would be a failure does it make sense in this case we were looking for exactly popular service terminology exactly 501 successes business make sense in this case we're looking for 501 or more successful successes but it was out of how many tries that will be our trials so you can we fixed that we had to fix that somewhere so that we can work with a problem so number one you have to have a fixed number of trials you can't just flip the coin forever must be a fixed number of trials number two all this one's got to be there you're gonna be happy about this one trust me do you remember hope well you don't have an addition rule we just found that out right you're all gonna be looking another video I'm have like a thousand to use tonight about on that video for addition rule but if you remember anything about the additional that you looked back at your notes which you should do occasionally I hope a long time between tests when you look back at their you notice there was really it mattered whether one probability depended on the other one didn't it it matters sometimes and we had to have it so that those things were independent otherwise we had a carryover we had a crossover we had to subtract that out remember that when they can both occur at the same time or where one probably affected the other one that that would make a big difference if we're having several probabilities in the road that would make a difference 46 let me explain in the court of this this one not so much exactly 501 that there's just one probability but this one 501 or more if you are flipping a coin and what you get the first time affects what you get the next time adding all these probabilities that could be extremely hard to do right because 501 getting ahead would affect the 502nd for the 503rd or the 504th and you have a huge massive thing to deal with you see we're talking about so these probabilities have to be completely independent not based on each other which means I flip the coin everything resets I flip coin everything resets back to 50/50 every single time I roll the die I pick it up back to 1/6 chance for all those sides you with me on this folks they've got to be independent they cannot the probability one one outcome cannot affect the probability of another or I should say the occurrence of one outcome does not affect the others trials must be independent that means the outcome of one trial does not affect any of the others third this is where we have the binomial part we just spoke about this each trial has to have only two outcomes either a success or a failure or what you qualify a success with you categorize a failure so each trial has only two others well that's going to naturally be there that's how we started something out and number four the probability of a success has to say the same for each trial that you do for instance if we're flipping a coin a thousand times we're not going to go halfway through and say oh we're going to switch coins and this one has more of a chance of beating heads get that that wouldn't work so long for us so the probability of getting a head has to be the same every time if we're rolling a die the probability of rolling a 3 if that was our success would have to be the same every single time it can't change so we could do this with the way to die and we just have to be that the probabilities don't change halfway through my experiment or a quarter like you or vary by traveler trial are you within that yes okay so the every time you repeat the trial the probability of a success is the same okay there is some notation that I got to give you when we're dealing that a binomial probability distribution we got some letters we got dealing first one is and little n little n usually stands for the number of things right the number in a sample in our case n stands for the number of trials working I'm gonna do this stem gears well kind of refer you back to this example and so I left on the board but if if we're doing this example what is our n in this case that's how many times we're repeating the trial sort and it would certainly be 1,000 very good if you're going back any notes maybe write this under under this one rewrite this example don't get those two confuses just rewrite that or what we're gonna do several more examples so just kind of keep this one your mind right now they trust me we'll do more examples I don't want you going back that the devil's not the idea that section 5.2 is a 5.3 idea so number trials end would be one thousand in this case that's how many times repeating that procedure there's a couple other letters one of them is little peanut big P little P this lowercase the P stands for the probability that you are going to get a successful trial it's little P going to change throughout your procedure that's what we said right here probabilities accessories are saying all trials so little P is going to be the probability of getting a successful outcome in each individual trial the probability of a successful outcome for each trial so put it in a single trial the probability of a successful outcome in a single trial okay what's the opposite of the letter P Q obviously it's written backwards right okay if little P stands for the probability of a successful trial for the probability of a successful outcome in a single trial the probability of QT is for what you think appropriate failure for sure this is the probability of a failing outcome in a single trial well we have a deal with our variable yet we have n and it's gonna be give it to you it's a number of trials that you have P lowercase the rupee is a presidency so while you'll also be given that the same acute not given those things also in what you're looking for is the X in our case the X is the number of successes that you're looking for okay the number of successes that you're looking for or the number of successes that occur in ten trials one last one big letter P it's always went to of X P of X what's that big letter P stand for its probability what's the X stand for again it's a probability of getting that exact number of successes okay this is a probability of getting this many successes we're gonna fill all the rest of this out we're gonna identify these items in just a second but before we do I need to make a couple things really clear for you so put your pencils down or they stop or pause for a second it's business when we're doing these things a lot of students get really confused some of this terminology and usually people are pretty good at the end and just and ever times repeating something but they get really confused between this one this one and this one these are tied together this is a probability shirt but this is just a number okay here's the difference between likes X is the number of successes you want P lowercase letter P it's a probability of each of those successes happening the probability of one single success occurring in your output in your trial that's what that means what this probability needs is the probability of getting that many successes so is XA probability it's exit probability X is just a number of successes you're looking for whatever that happens to be X is the number of successes you're looking for is this a probability that's the probability of each one of those successes is to have successes each one those successes is happening this is this a probability definitely this the probability that you're going to get all of these successes exactly that amount of successes do you see the interplay between minute these are both probabilities this is trial by trial this is overall which is based on your memories just azuma before okay so this probability of each success that you get this the number of successes you're looking for this is the probability of getting that number of successes so let's go ahead and fill out the rest of the stuff we knew n was a thousand we're going to look for P little P little P is a probability you're going to get a successful outcome now in our case what is our success in our case what was it get the head then is a success simply please determines a success is not getting 501 heads a success is not hitting 500 once it hits a success in this procedure is getting a single head that is a success success is getting paid ahead just one why do I say that well because when we look at s ex counsel knows the difference here ex counts up the number of successes worth looking for the number of successes were looking for how many successes are we looking for in this in this example how many successes right here we know difference the success the success isn't getting off isn't giving 501 heads the success is getting one head we're just looking for that to happen how many times times that's where X comes in so success is getting a head X 501 successes that's how that plays in there were five didn't want successes what's the probability of getting one success flipping a coin so assume it's a standard coin flipping a coin is notice how you have to identify success as a single look at the board no say you have to identify the success as a single because this probability is based on that one single enough do you have to do that X's you know that's that's a number of success at Pease point five zero I'm just cute yeah why one more thing the last thing we'll talk about how much does P plus Q for sure probably a success or failure willing to up concern it's gotta be one there's also these two P equals one minus QB subtracted or Q equals or Q equals one minus the only thing we haven't done is figure out this but we're gonna go ahead and figure that one out next time so we now know n we know probability success for single trial public failure for single trial I success is a single outcome it's a single outcome X gives you the number of successes that you were looking for don't worry about it so we're trying to figure out this these binomial terms basically is what I want you to understand right now what is our n our lower case P or Q or X the billion x will deal with in just a second so we're looking at binomial probabilities we're realizing that binomial means either success or failure and we need to identify all of these letters so and which we just talked about was the number of trials that were we're having to occur in our whole entire procedure here so how many trials are we dealing with in this particular way to die problem and we're rolling the die ten times so this trial is being repeated in 10 times so yeah and this time cursory Anniston let's go ahead and let's look for the probability well firstly we need to identify something we need to identify what is our success and what we're considering to be a failure I want you to identify these even if your problem doesn't ask you for it identify those on your problems on the test or on your homework be identifying that I'll ask you for one test I want you to understand what a success is in this case and what a failure is in this case so let's look at this the probability of rolling a four is 30% but dies well ten times find the probability of one exactly eight fours what is a success in this okay I thought it would be confused here is the success rolling eight for spores the assistant is a successful in a form for a success has to be trialed by trial based otherwise you cannot find your lowercase letter P it's got to be for one trial so you know you can't find that the whole idea is gonna be defined the probability of a certain number of successes that's based on both case letter P which is an individual by individual trial success rate or probability steps so in our case a success is it bullying eight force it's really just one form we're going to be looking for a certain number of successes do you see that they're gonna play there so in our case they success is just will any form really a formula okay what's a failure what would that be sure we have we have there were six outcomes that are possible so if you roll a one two three five or six even though those are all different outcomes right there's six points love comes notice that this is still a binomial probability because we have only one success in the rest of players we say four is our success that's why we have this in South ruling a single for that's a success everything else would be a failure so in our case failure is rolling anything else only one two three five and six feel okay with n the success and failures okay I want you to be identifying all that stuff on your own as well let's go ahead looks and look for the probability of success that's our lowercase letter P the probability of success notice how if if I have the success is willing 8/4 I can't even find that right now you have no idea that's what the question was actually asking you so these things have to work together a success should be probably a success right here so what's the problem is a single success a rolling importance is up there at the very beginning probably only four is oh so this is the way to die so it's not just automatically when I have six you got to read for what the probability of success actually is so if our success is rolling a4 what you should say up there somewhere should be given to you in some way what the probability of success actually is so probably going for 30% am I gonna put 30 what am I going to put we're know probabilities are between zero and one so if I give you a percentage we got translate that through them doesn't it so we have that let's shut up our cellphones pieces if P is point three zero can you find Q can you find Q the probability of failure these things are complementary you're either a success or a failure our son you're successful you're a failure no middle ground here is average now well if Europe they do probability of beginning a success it's 30% of point three zero the probability or is everything else so what is that yeah the way we're figuring out the point seven zero we're taking 1 minus P that's what that is 1 minus the probability success it gives you the probability of failure those things have to add to 1 therefore the number 1 minus either with probably success or brother that will give you the ultimate what else we'll just oh yeah thanks X what is X stand for again number of successes we're looking for so notice how a success would be a single thing the number of successes how many of these things how many successes I'm looking for in my whole entire procedure so how many successes and by looking for best winning it does accessible and for how many force were looking for very good yeah eight for eight times that we're successful looking for exactly eight times we've rolled that poor in this case that stuff now comes the question how do we find the probability how do we find the probability of exactly eight successes probability of success I want you to notice how the four is not here we're not finding the probability of a four or honest is kinda weird but four is going to be irrelevant what this comes down to is the number of successes you're looking for out of a certain number of trials the fact that we roll enforce that's all great but we're actually just looking for the number of successes we did exactly the same thing if we had like a way to die and I said the probability of getting a head is thirty percent probably gave a tail is seven percent we're looking to flip the coin 10 times and get eight heads that's exactly the same problem exactly the same problem so this is all based on the number of successes compared to the number of trials and considering the probability of the success and failure for each trial you can see what I mean in just a little bit okay just a little bit so we're looking for the probability of exactly eight successes oh my gosh how the world do we knew that olders a formula there's a formula for how to find probabilities in the binomial situation which we have here ready for the formula of course you are I told you I'm on today I told you what my my math injury so our formula here's the formula for the binomial probability formula if you want to find the probability of a certain number of successes the probability of a certain number of successes here's how the formula looks I'm going to give you the formula they're gonna tell you why it works the way that's silly that looks familiar that look that should look familiar to you you just did that in Section 4.7 what is this formula right now it's either a permutation combination it's one of these similar to the combination it is the combination oh you know what in fact we make that I think I actually put it or it should be nice oh yeah now that makes a lot more sense so now they look familiar to you did it I've never seen that yep this is now the the combination form in this is this would be + C X that's what this is that's what this section is it's a number of combinations of ways that we can accomplish eight successes or how many successes out of 10 trials that many many combinations is what we're looking for so we're taking each of those combinations we're not done with the form of though what we do now is multiply all those successes times the probability of success for each one that means to the X power so X right here is the number of successes we're looking for we identify this is probability of X that's the number of successes this is every time we have a success we have to multiply that probability that's what the P of X comes from now we also have to multiply every failure types of probability of failure because we're looking for an exact number here so this is kind of interesting to think about with in our situation we actually want eight successes and we actually want how many failures there were two failures we want two failures in order to get those eight successes are you within them so in our case we have to multiply not only by the puggly of success a certain number of times we also want the probability of failure a certain number of times what does this have to be do you think if we have X successes we want how many failures in this case two failures sure but how about in variable say that again a minus two for this situation sure that would be ten minus two I don't have a two here I have a I'm sorry not a minus 2 we got eight months eight actually that would never close and - what if you have X successes and you have n total trials how many failures do you have we have the total number of trials minus the number of successes that gives you your failures if you have eight successes out of ten trials we want to or 10 minus 8 failures you with me on this folks yes or no so this is the number of combinations that we can accomplish X successes this is the combinations of that happening where you have different like a different scenario playing out on roll your die because you're not just going to take your die and roll let's see we're doing pores before before before for eight times and then whatever else than the next two we can roll a four and then a two and then four or four or four and then it's seven or nine 750ml 7 and the 6 and then a couple more fours we could do that or we could roll a four and then a three and then some other ones there were four and then a five and some other ones we can do all those situations all those play into the number of ways we can get eight fours and to something else's yes do something on whatever they are real character on the same category of billion we take each of those successes we multiply that times the probability of P the probability of the successes that's where this P of X is coming from this is over and over again if we had eight successes we doing ugly success eight times that's where we get the exponent probably the vinegar we're taking well the total number trials - that many successes gives us the number of failures and we want to feathers you multiply this twice times itself properly family twice that's where this formula is coming from now let's see if we can use this formula to find this probability so we want the probability oh by the way if you want to make this a little more concise all that why don't instead of doing all this work because your calculator will actually do that won't it let's just call it that let's do n CX times P of x times Q n minus X that's a little bit easier to accomplish on your calculator we know how to do this that's just an exponent that's just an exponent I'm going to do that over there so we can seem a little bit better right now we're going to look for the probability of eight successes the probability of eight successes what this says to do is first off we're gonna have n CX what was our n in this case yo you're also okay on that right then we have ten okay let's do this we got ten see what was the next if X is ncx in our case we have 10 trials we're looking for eight exactly eight successes x we're going to take a probability of each of those successes to the power which happens to also be the number of successes so what's our lowercase letter P what would go right here I'm going to put that in parenthesis it looks a little bit more nice that way so you know that's a decimal in their point three zero what power is that going to be raised to lace gentle sure Y to the eighth power again that's our number of successes I want you to think about what we're doing think what you're doing this is the probability of each success we're multiplying that probability of each success times itself for how many successes we want so 0.3 one times point three times 0.38 times that's where that eights coming from we want eight successes multiply the probabilities excess time to sell many times that's point three zero two to eight hours of sleep in the case next up we take the probability of failure what's that probability and we're gonna take that to what power in this case yeah 10 minus 8 which is gonna give us that - so endless 10x was 8 so our probability is NC eight times point three zero to the eighth power times point seven zero to the second power that will give us a probability getting exactly exactly eight successes again the reason why this formula works this is every possible way you can get eight successes out of ten bullets that's what that says this is a probability success we want eight of them it's probably the failure we want two of them you will find those brothers together Malaysian rule and you have this formula so basic kind of a multiplication well how this is right let's do it find ten eight you should know how to do that in your calculator we've done that a couple times right before back what is it only 45 so there's 45 ladies growing a diet that control it ten times and get exactly eight four so four oh by the way I was gonna mention us put the board roughly the number four was the fort was the number actually trying to roll wasn't it did you use the number four at all that's not trying to tell you is it this is not based on a specific value that you're trying to get what is based on is the number of successes you're looking at considering your probabilities do you see the difference there for doesn't even play into our equation at all it's just about the number of successes compared to the probability of each success from believe each failure that number of trials you have the number successes you're looking for doesn't have to do with the four it's all about the number of successes yes no that's what I was trying to say earlier what I was saying so 45 that's this little part times let's do point three zero to the eighth power six point five six okay times ten negative five right because when you multiply a decimal times decimal you actually get a smaller decimal so the candy see you put six point five six you're way off and you're way way off you're gonna get a number bigger than one you have a probability of like oh I don't know 250 or something then your probability has to be between zero and one so when you read your calculator you get times 10 to the negative fifth you have to know what to do with that so tell me exactly what you got six point five six is that what it says okay six point five six one times 10 to the negative do you know what that means that means you are one two three four five decimal places this way that's what that means so you do not put six point five six one you put point zero zero zero zero six five six one should you round these numbers do you think man they're so small if you round them you're gonna be off all right you're gonna be pretty off on that so we can't round them we're gonna put exactly what our calculator says to the end and we're gonna do the same thing point seven zero squared that should give you point four nine so 45 times point zero zero zero zero six five six one times 0.49 okay time yeah okay so that's that's good enough for us in fact most of the time we're gonna end right about here those summer end right about there but to get right about here or here here you have to be very accurate with your numbers you cannot round those things remember the rounding I told you about don't round till the very last step that's what we're doing here so what does that even mean oh my if you will it die at ten times what is the probability that you are going to get exactly eight number fours using this information is it it's like point one percent 0.14 percent that's pretty rare like pretty pretty rare that you're going to roll that die ten times and get eight fours and then two of something else that's pretty rare by the way what distinguishes rare from Unruh or usual versus unusual what what number really before probability speaking ones with zero five nine point five W fifty percent eight point zero five that's five percent if this is less than five percent which it definitely is then this is considered unusual an event and this is very very unusual for the very unusual there now let's consider that we did this how the next question the next question is what's the probability of rolling at most eight fours in this class you're going to be very good at knowing the interplay between at most at least more than or less than and none those four situations occur a lot more than less than at most at least and that none is the easy one what about finding the probability using the same information of rolling at most eight force probably going at most eight fours let's think about we're stupid rule ten times okay same exact information up here same execu mission you can understand though what at most means if you have it most eight dollars in your pocket at most eight dollars how much money could you potentially have in your pocket you couldn't cuz you have eight dollars could you have nine dollars no but you could have eight right cuz you have seven six but no you can't have negative but you can have in order to satisfy this 0 you can see row dollars right that's it most it's not even anything have one two three four five six seven or you could have eight you can get you could get up to eight including eight and that would satisfy at most does that make sense less than what if I said less than eight less and less than eight would it be included in that there would be seven six five four three two one or zero if I said at least eight at least eight that would be eight included and then one nine nine ten or more than that if I said more than eight more than eight would that be included in more than eight when it concluded that if you have more than eight dollars do you have eight dollars that's the interplay between the at most at least more than less than you know you really know those problems right so we're looking at at most eight at most eight means you could have zero one or two or three or four or five or six or seven or eight number fours that come up in your 10 rolls are you with this so this is kind of interesting we just said that zero would satisfy this right zero successes zero successes with says fence-mending successes here if I get no force is that at most eight yes we also said or you can get one for wood one for one success wood one success one for satisfy at most eight or you can have two successes when two successes - for to satisfy endless eight you with minutes or three or four or five or six or seven or eight do you agree with me that any of these situations would satisfy this situation either zero or one or two or three point four or five or six or seven or eight to over the board we talk about the order hopefully you didn't go back to the freshmen ever on what or means how do I calculate a probability before what do I do with those yeah this is a probability of zero occurring or one occurring or two or three or four or five or six or seven for me you add all those probabilities together because that's an addition rule this is an invisible or disjoint sets you can't have both three successes and four successes can you can you get both three fours and fours when you're rolling the die ten times can you get a certain number of successes and a different number of successes for instance imagine your head you're willing to die ten times you're rolling the die can you get out of your ten rolls can you get both three fours only exactly three fours and four fours at the same can you get both of those exactly three fours and exactly four fours those cannot half at the same time you can't you can't get that so here when we're doing this these are disjoint sets disjoint outcomes we add them all up we don't have to worry about a crossover that's what I'm trying to say here there's no crossover there's no end of them happen at the same time we just add up into those probabilities so the probability growing at most eight fours comes down to the probability of only zero plus the probability of only one plus two plus three plus four five six seven and eight how do we find each of these probabilities it's that okay can't you find the probability of zero let's do this real quick let's see what the probability 0 would be probability of 0 is that rolling the 0 descending only to 0 close what does the probability of 0 mean now a force means if you got 0 successes it's not the probability of rolling a 1 I don't care about that what care about us is 1 success there wasn't the profitable 1/8 that was definitely not maybe we can't even with me this is the probability of going 8-4 or getting a success and some people understand the difference between those things you kind of have to really get this stuff otherwise just be asking a question or kind of think about this a little bit so this is not the probability of going a zero that can't even happen all we can get is one through six this is not the probability going one don't care about that it's not the probability of rolling a four I care about how many fours I'm rolling this is the probability of rolling zero forced out of ten this is a probability of only one four out of ten this is probably going to force out of ten or eight four out of ten do you see the difference there imagine if you do okay that's that's the big point here so the probability of rolling zero four is out of 10 or X become zero sure with our end chain on what our end change what an exchange that we got becomes zero what our peach ange excellent becomes zero q would become would change and minus X that would become 10 this would be zero that would be 10 this would be zero and choose zero would still be something that you need to calculate it was still happening calculator something can become zero this probably does exist you not do that for this one then you'd have to do that for this one and do that eight times add them all up do you wanna do that why not did this was exist I give all my saying so yeah it takes way too long so you could do it though couldn't you here's one here's one two three four five six seven eight that's nine formulas that you would have to do nine times each time the only thing that will be changing is X and you redo your formula find out this number nine times then add them all up however some guy has a lot of time on his hands it was me and has done this for you you'll see what it's what's done for you that's kind of Awesome do you remember how I told you that this action this whole act of doing this really had nothing to do with the fact that we were only made for it had to do with the fact that we had ten trials and we're trying to find eight successes with a certain probability and you agreed with that and you said okay it's really nothing dude before that's all there were successes something every ties then and you made a chart basically so in the book that magically appeared my hands weren't you gonna find on page register 749 show and tell 7:49 you get this nice table thing looks like that a list of the very toughest binomial probabilities never talk about binomial probabilities just right now binomial probabilities some bag did all your work for you he found all the probabilities to the fourth decimal place third displace sorry of all the situations that will accomplish or that will have to face in this so here's Senator Reid your table here if you read your table says binomial probabilities on the top I want you to notice up at the top left-hand it says some letters we should be familiar with what do you see I see em n stands for what X stands for successes for sure it's not based on the value you're trying to get some number you're trying to get also the letter what's that letter yeah can you read that is your eyesight that good that's a that's a P once you turn those lights off for me would you that's the the P that's a probability of getting is this the probability of getting two successes or is this the profit of each individual success that's kind of cool it does the work for you so this is the probability of getting each individual success based on this information here's your end here's the number successes you're looking for now you'll notice that if you will have two trials there's only three situations you get out 0 successes 1 successes or 2 successes that's it same thing with 3 trials you have 0 1 2 or 3 it builds as you keep going through and going through so let's see if we can do our problem I hope you remember the probability over here was point 0 0 1 4 4 right let's go ahead and try to find our situation on this on this table over here what was our end in our role in the die situation let's see go we have unseen here stand and is nine right down here and it's tenth so we should be in this this same rhythm here we're in is ten are you with me folks next thing we're going to do is we're going to look for the number of successes that we were trying to find now in our situation we were looking for how many successes exactly eight successes that's that's zero we're eight successes and the probability of each success was how much thirty percent okay so what we're gonna do probably successful finding that we're finding and is ten eight successes we're looking for let's let's see what those things cross point zero zero one we were more accurate one week zero point zero zero one four but that's that's the same value that's what the guy found he just rounded to three decimal places so we can find that value point zeros or one just by looking to the table now why aren't why we're doing this we could just do the formula well when we are looking for the probability of at most well jeez at most a battement zero or one or two or three and we would have to do the formula nine times to find out at most eight successes remember that can you find the probability of zero successes just struggling but look at this it's point zero two eight that's great how is the probability of one success oh it's one point two one so it's already to the net 42 successes 0.233 three successes point to sixty-seven how can I find the probability of at most eight successes what am I going to do with those values I was just reading to you yeah let's just add them we know that this was addition rule right so I what we had up here on the board was the probability of 0 plus probability of 1 plus rather than two successes plus three four five six seven eight let's just add from here down to and include here's what you have to be good at okay everybody's going to know to do this it's not hard you just go to your end that the X is listed for you I mean even letters are the piece listed for you you just have to know what at most means at least means more than and less than you have to know basically whether to include this eight or not whether the ad of all these or all these that's what you have to know also one little piece of information the zero plus just use zero from that zero plus means less than one thousandth so point zero zero zero of something that's what that means so you would be able to find that on a calculator this won't let you do that so you you the fu zero for that so for us let's go ahead and do that let's write these down point zero two eighty point one to one point two thirty three point two six seven point two zero zero and so on until you to the eight okay write those down because I'm going to I'm going to show you how to do this on a calculator this is good yeah you should have like 0.9999 good Djamel is written down does anybody have votes are down okay so hopefully we have those written down if not we'll do it like okay now after people there you go your scientific calculator will not do this with this fantastic detective what you're going to do is turn on the calculator we are dealing with the binomial what's it called possibility formula and it's based on the probability distribution form under grade distribution so look real careful you should have a distribution button this that this true we're going to go to this to go to the distributions right above the variables button right there it pulls that up on the screen now look eventually we will actually be dealing with all of this stuff right here maybe even this stuff but scroll down a little bit would you scroll down a little bit and you're gonna see that numbers 9 I'm sorry and a those say binomial PDF and binomialcdf don't go into the poison you know the only poison that's Boise on actually but don't go to that one you'll get sick we're just gonna go in here you go to poison you got before I go to binomial PDF and binomial CDF this is a point probability this is a cumulative probability here's how this works watch carefully if you want to find out an exact value an exact number successes that's considered a point probability that's gonna be your PDF this is what's so cool not your calculator if you want to find out up to and including something that's cumulative so if I want to find any most something that's up to and including eight even know that if I wanted to find that slam ideal up to but not including so I don't want to plunge it not ain't here but seven one less than that does that make sense to you so you have to be smarter than your calculator is you get a bunch of the right numbers but it will do the math for you yourself what you're going to do let's try the exactly eight first exactly eight you're going to punch in the number of trials you have first so right now the number of trials is how many then put a comma then put it the probability of success for those trials in our case it's a point three zero comma then you put point three zero that's a probability of success and then you put how many successes you want so comma that's right here I want how many successes you don't have to even put the and procedures prison in turn point zero zero one four four six seven that's exactly what we discovered with all that work with the formulas now if you want to find it up to and including okay up to and including I have to do is kind of look we've run out of time I'm gonna go video go to distribution again go down to binomial CDF press Enter what this will do you have the same information but what this will do it will add up all the probabilities up to a certain number for you and including that number so you have to be good at knowing whether you chose to include that number or not because if you put it it's going to include it if you want less than something you go one less than that number so we'd go ten trials probability success it's still point three zero we want is what you're saying here folks you said I went up to and including eight successes point nine nine nine eight five that just added everything up to you and it's more accurate than your table how many will understood that okay watch that video again if you didn't quite catch how to do that in calculator this is very very useful for you alright plus the probability getting seven parts what accomplished my game if I get five marks six or seven parts I win true if I pull up seven parts in a row do I win the game pull it six hearts do it win the game well the five parts we win the game how about four marks now I feel good so those right there this right here this would be the only ways that we can accomplish whatever game this is the probability of getting at least five parts that's the worst part I've ever gone I didn't know what that looks like fishtail and last time I showed you a couple ways to go ahead and find this without doing a whole lot of work because honestly you're not gonna want to do the formula three times are you woman that's kind of kind of horrible I mean it's not bad for finding one but it's kind of horrible to find more than more than even one takes a long time you have a table that works for you know seven calculator that works for you before we go ahead and do that we'll show you how to find all those things I want to determine a couple other other ideas we're gonna change our game a little bit okay firstly let's say that this is this is our first game we're gonna figure this out in just a bit let's say that the next game is you win the game if you get exactly 4 hearts exactly four marks how many successes are we looking for there how many successes from in four parts one success how many parts my success as a heart right how many parts am I looking for so before successes they tell me something what five hearts work for this you lose with three hearts word for that's you tha's okay so in this situation you would have to draw exactly four no more no less is that hard to do do you think it have to be like one of these combinations no hard yes art yes or no yes art yes or no more that be one combination that's hard to do right the every every every combination would would have to have only four hearts in it and everything else would have to be a different scene so that's one more game I want to do this what's the probability let's say this is our game now our game is you can have at most three hearts at most three so let's see how that game would work out if you drew a card at most three could you draw no hearts at which how would you win that game if you could get at most think about this is I'm writing down how would you win this way if you only could win if you got that most no hearts work sure what else would work one more to Mars win three parts work that's at most three how about four months five six seven elevate would even consider eight no because it's outside of our range more than how about this you have to win the game in order to win the game you have to have more than five parts more using five let's say you have to have more than two hearts in order where the game games have to have more than two parts give me this scenarios for winning this game more than two hearts because there is for winning three okay alle to know why not - I need more than two more than two so three four five six seven ok one more in order to win the game you need less than 60 less than six months oh that's one of the better ones and it was gone that's pretty good all right less than six Hertz or when they gave me less than six hearts so let's look at that what wins for less than six parts zero wins this lesson six one two three four five six no that's not less than six these are most of the scenarios you're going to see most things we see are at least exactly at most more than less than those are the scenarios so now we have one two three four we have five games we're gonna play on this with this one example okay so I have your cards we're drawing one putting a package drawn one put the back drawn one put it back we're gonna do that seven times what I want to do is find the probability of getting at least five hearts then exactly four at most three more than two less than six are all the problems gonna be the same so the chance of winner you'd be different for every single game that we play I'm gonna show this to you on the table we're also gonna do this on the calculator did you guys bring your tables today yeah you should probably bring them so you know what we'll be doing this from time to time on your tables so for now we're good look at the table that I have coming up here in just a bit also 200 calculators will then get your graphic collectors out if you have them okay yeah I'll give you the I'll give you this table the proper table for your test when the time comes to it so that you have at least the information there so I can't say like yeah memorize this whole thing hahahahaha that would take your life so let's look at the probability of finding at least five parts you have to memorize those by the way or have these written down because we're going to do this on the board of world the white screen actually can you see it when it comes up do you remember how to use this table what is this poem for and so in our case in our game what n are we using feminine so we're gonna be down here it's below our screen right now going to move this up what's the X stand for number one no successes successes this is my successes what's this stand for probably for each success notice out that has to be based on success by success from travellish basis so what is our probability here well wait a second what are we gonna do it bad it's right in the middle of that this table isn't gonna work exactly perfect for us in this case you're gonna have to use your calculator use the formula or you're gonna have to guesstimate there which is not going to be the best for us so just to make this a little bit better can I can I pledge that for a second so you can use that I'm going to fudge it for a second so I'm probably leaving a little inaccurate let's say that we have point two zero so I'm lying here yes I know I'm lying but I want you to be able to use that table and get some appropriate answers know that your calculator by the way we'll use the point two five just fine with no problems whatsoever the table obviously has some drawbacks to it we can do the 1 5 10 percent 20 percent but then it goes up every 10 percent after that and tell you to the 95 and 99 so this range we'd have to go right between those just guesstimate between our point two hundred point three wouldn't be the best case for us you guys with me on that so all you can expect on your tests on your homework they might give you some of that on the test you can expect that one of your probabilities will be in one of these homes okey-dokey so we're gonna fudge this a little bit I know that this is not accurate but we're gonna say it's point two zero let's find the probability of at least five hearts if we do at least five parts notice we're going to be in the one to three we're going to be in the fourth column the whole time are you with me on this yeah of course I do fortunately for us the hue is really not on this table so if I did then yeah that would be a point eight zero now so let's check that out I've moved it so that this is our fourth column you know that this is a point to zero what we're looking for is right here the probability of at least five parts happening what accomplished a probability of five marks we already went through this we could get five six or seven five six or seven those three situations let's look at that here's five parts this is a probability of exactly five parts coming out of this situation here's six parts and here's seven parts so if we added all that together remember we treat this is pretty much a zero this is less than point zero zero one so this is going to be how much altogether what you let you say point zero zero 1 plus 0 plus plus zero plus beginning at point zero zero so the probability that you're gonna win my game is it good or not for this particular one getting at least five bars is that good why is it unusual yeah it's not above point zero five so it's less than or equal to point zero five so that would be very unusual to win this game oh it's a big mistake yeah it doesn't happen often does it but it does yeah just kidding folks I made a mistake it did the point zero one I should have been point zero four yeah point zero zero four plus zero plus zero point zero zero four so the probability is still not very good I mean we needed this to be a five right there this is way less than that so this is an unusual unusual win if you were doing this it would not be normal vision if you'll okay with it so far okay good let's find the probability of getting exactly four hearts exactly four parts so let's look at that exactly four so I see four right here I know I'm in this column do I need to add anything on to this number if I'm looking for it exactly four horse so this is point zero to nine which game would you rather play getting it at least five parts or getting exactly four words so if you get to choose between those games you'd probably say I'd rather play the getting exactly four parts even though the probability still sucks of you winning it's still unusual in it less than 5% still unusual it's better than this game let's look at at most three parts at most three parts if you had at most three we already said that was what was the most three again zero work will us work three is included in that at most three so we'd go back over to our calls we go okay zero hearts works okay one heart works two marks work all the way to three so we're looking at those four numbers right there zero one two or three inclusive so we're gonna add all those things up so point two one zero point three six seven point two seven five point one one five can someone have those for me tell me what you get out of that right now much point nine seven hey which game would you rather play now huh once again once you rather play you're almost guaranteed to win right you get at most three hearts it's almost guaranteed a win now of course I did I pledged the probabilities a little bit so we can use our table accurately but still that's a really high probability I'm getting at most three points okay let's look at more than two hearts what's the problem in getting more than two hearts what's more than two mean it's two included not and that's basically the whole idea here more than two so more than two we'd have to go here's to do I include that number or not but I want more than two so I do this one this one this one this one this one I do the rest of them so three or more so what you understand is that more than two is the same thing as three or more so you guys getting that also the same as at least three so Adam that column we do the point one one five point zero two nine point zero four and then these are considered zero how much is that what wolf or a so we added them how about them less than six months less than six parts here what counts as less than six parts five so six is not included so we'd have to do five or less so less than six is five or less or at most five it's the same thing if you add all that up how much are you going to get should be pretty close to a hundred percent eight point nine nine nine wouldn't exactly one okay so this is a little bit off because this fish cannot be won but it will round to one there is some probability here so I'm not going to put I'm not put one for you if you ever get this this case and you add up and you haven't added the whole column here's what you know about this the whole column will equal one there's one hundred percent probability you're gonna land in one of these cases however there is some probability here it's just rounded in this particular case these round in such a way that when you add them up you get one point zero zero right so if you ever get that you haven't added the whole column don't do one point zero zero just do like 0.9999 give me an extra nine on there I don't know this 9 is gonna be accurate but it's definitely not one point zero zero I need you to know that what you raishin have you okay with this so far now would you like to see the way to do this on the calculator okay let's do that so you see okay now I kind of want to keep the zoomed in so you'll have to kind of you want to remember what buttons that we are pressing firstly where did we go to deal with the binomial distribution not math we're now in can you see it down there corner distribution so we're going to use second that's where your suppressed and distribution is right there by my fingers so we're going to press the VAR s button that brings you to your distribution there's better so we're looking at our distribution is distribution are we gonna go to that's way down here paths are chi squared plus RT that's our app down here to buy them you have two choices we give the PDF and the CDF which one of them works for a single value the P or the see which one the piece for a point one single point the C means cumulative add that will add them all up for you you put in the information the same way so you really need to know which case you are working on in our case for at least five parts at least five parts what's gonna be working on a we working on a C or a P and here's the problem with this negative come back to this one I want to work on a couple of these other ones first okay I'll show you then at least I'll show you the more than and just a little bit I want to show you the exactly first then I'm going to go through the at most the less than then we'll do those other two so let's work on exactly for right now for it we're gonna be on the P of the CP okay for and most of them be on the pier to see see for the rest will be on the scene will be all on the cumulative because we're talking about it most less than more than all that stuff and it says exactly it's the P if it's anything else it's the C not you may be okay with that let's do the exactly for for exactly for do you remember what we plug in first so now I'm going up to the PMI highlight the zero I'll press ENTER that should be on our screen right there what do we plug in first for that the not the X it's the 77 what's in what's what's the n stand for n set for seven it's the seven that's our end then we find our problem and we put in what next notice you put in they're probably literature for the probability probably what we put in this right there or if you want to do the previous example you can do that right now you put in point two five right now now we're reform as where we did this so we could use the table of course the realistic example would be point two five you would be on that I want to show that to you so you know that your table doesn't work perfect in every case it doesn't have everything if this is a point two eight you certainly would be able to use the table at all I'd be very it'd be awful so here are point two zero we're going to use this so you can see the the numbers do match up or pretty close to match up on this so point to 0 that is our probability success for each trial what's the next thing that we plug in now that comes our X so with the calculator you do at the end then your little kiss letter P and then whatever actually could for in our case what's our exhalation gentlemen we're over there four point zero two eight six point zero two eight six it's impossible for you to turn off with one of those lights so in the calculator we got point zero two eight seven are these two numbers the same this one's actually supposed to be this one it's just rounded it's rounded more to fit on the table are you guys okay so far all right let's do the at most three parts if you have that in most three parts stick with me here relation gentlemen you don't have a cover there's good refresher method on how to do this at most three parts are we gonna be in the P or the C once again see so go back to my distribution I'll go down to the binomial CDF will press ENTER on the CDF that's the a there are trials change there are files change so seven you know our probability of success for each trial change we still point two zero one two now then the X's change now you need to know what it most three means is when I'm talking about do we want to go up to three non-inclusive or up to three including three listen if you want to include the three equal to three if you don't want to include the three you put the tooth does that make sense so we want it let's say where we're at most three Emmy zero one two or the three we're gonna put the three this is going to add up to two three including the three and it's gonna be point nine six six six does that round to 0.96 7 linka tables of Genesis okay so get 616 forever let's do the less than six hearts let's spend six hearts what works for less than six hearts does the six work or not that's the six work for less than six less and gently did you with me mrs. six work know what works five or less one two three four or five zero one two three four five so when you do this one you got to go back to your distribution every time we're still on the cdf but i want you to watch what i'm going to do here if we're going to do less than six be smarter than your calculator if your customer doesn't really know what you mean it just says I know buggy number seven you get any smarter than than the problem in your calculator you got to know that what I'm really talking about is five or less including the five it's less than six is five or less those are the same thing so I'm gonna put it in on six because that would add the six I'm gonna put it in five because I know that really the only way that I can accomplish this is 0 1 2 3 4 or 5 I put in the 5 there I press ENTER 0.999 6 of the law six according to calculus what we get okay these are in half your own can't find what we have just found so far cool now the next ones we're going to do which ones have we done the more than two hearts for the at least five hearts around here's the deal it's oftentimes a lot easier to find out a complement of this thing because your calculator won't directly give you this if you plug in two you know what it's gonna give you zero one or two you've seen that right that's what these give you it's gonna be zero one or two if you plug it once and give you zero one people again three it's not gonna give you greater than that seem you be zero one two or three so here's the idea in order to use your calculator effectively you need to understand the complement of what this is so you understand that if I want more than two hearts what we should be doing is finding the probability what's the opposite of more than two parts more than two what's the opposite of more than two at most one or less than less than three probably at most two would be the best way to think of that so the complement of more than two hearts is at most why are we doing that well we know one thing about compliments compliments have to add up to what number let's just take this as one - at most two one - at most to check out why don't these things compliments more than two and at most two more than to think about more than two can you please think about more than two points what's more than to me three four but what's at most to me are those things compliment your either three four five six seven or zero one two those things are compliments instead of going for this one directly in your calculator you gotta take one minus the compliment this would be our complimentary so let's look for at most two subtract that from 1 1 we'll go to our distribution button we'll go down to the cumulative again we'll plug in the 7 will plug in the point 2 will plug in what's the next number we're gonna plug in 2 2 now when you play on that you remember what this is giving you right here when you do the the two that's giving you zero or one or two does that make sense to you it's not giving what you want it's giving you the compliment of what you want go press Enter let's give it as point eight five 125 to zero well just doing the math in my head are these things equal take your cup of if you don't believe me take one minus 0.85 to zero is it the same another history with me on that one considerably let's try one more attend over here in the dark change my size changement sites here we want the probability of at least five parts at least five parts means what what does at least five me - you have no support okay the company would be at most four but what's at least five mean does it include the fiber not yeah five six seven what's the opposite or complement of five six seven so we want at most opposite for the compliment of at least five parts is at most Forbes so if I want at least five parts I'm going to take one minus the probability that most sport works those are complementary events we know they have to add up to 1 so if I say properly at 0 1 2 3 or 4 subtract that probability for my 1 I get the probability of 5 6 or 7 but if you have your okay good all right let's go ahead and do that probability we'll go back to our distribution go down to our cumulative just like we've been doing well since around 7 or point to 0 and we're gonna go for now what number we're gonna plug in after that say 11 yeah we'll do it for we want up to it any including for 0.995 3 oops Oh point nine nine five three can't tell me what is one minus point nine nine five three please Emily point zero zero four seven hey why aren't these the same why isn't it that when you round this one it doesn't exactly equal this one like all the other ones why isn't that where these actually zeros so if you were to add those up you know what they're probably pretty close to point zero zero zero seven it's probably even pretty close to that that's why these things aren't exact as the table is approximating it's not going to more than three decimal places your calculator G is what you really have this example made sense to you good all right that's fantastic

Channel: Professor Leonard

Views: 221,282

Rating: 4.9508104 out of 5

Keywords: Professor, Leonard, Probability Distribution, Binomial Distribution (Literature Subject), Statistics (Field Of Study), Probability (Measurement System), Lecture (Type Of Public Presentation), Math

Id: iGKSxMGX0Do

Length: 92min 30sec (5550 seconds)

Published: Wed Dec 14 2011