The Binomial Experiment and the Binomial Formula (6.5)

in this video we'll be learning about the binomial setting and the binomial formula when we talk about the binomial probability distribution we are referring to the probability of a success or a failure in an experiment that is repeated multiple times you can easily remember this by paying attention to the prefix bi which literally means two for example in bicycles there are two wheels and in binoculars there are two lenses however in binomial probabilities there are two outcomes a success or a failure before we do some practice problems we need to talk about the binomial setting the binomial setting must satisfy four conditions the first condition is that the number of trials n must be fixed the second condition is that there are only two possible outcomes for each trial you can either have a success or a failure the third condition is that the probability of success must be constant for every trial and finally the fourth condition is that each trial must be independent this means that the outcome of one trial does not influence the outcome of another trial an experiment that satisfies these four conditions is called a binomial experiment the binomial setting will make more sense after we do an example so let's jump into it if you flip a regular coin three times what is the probability of getting exactly one head and is this a binomial experiment feel free to pause the video at this point so you can try this question for yourself since the question says that we are flipping the coin three times I will have three blank spaces one for each flip the first blank space is for the first flip the second is for the second flip and the third is for the third flip in order to solve this problem we have to realize that there are only three possible ways for us to get exactly one head the first way is to get heads on the first flip and then tails on the second and third flip the second way is to get tails on the first flip heads on the second flip and then tails on the third flip the third and final way is to get tails on the first and second flip and then getting heads on the third flip so overall we see that there are three different outcomes where we get exactly one head and we see that they vary based on the order now all we have to do is calculate the probabilities of each outcome and add them together to get the answer we know that the probability for getting heads is 0.5 and the probability of getting tails is also 0.5 to calculate the probability of the first outcome we will multiply the probability of heads times the probability of tails times the probability of tails again this is equal to 0.5 times 0.5 times 0.5 and this is equal to 0.125 when we calculate the probabilities for the second and third outcome you'll find that they will also be equal to 0.125 now all we have to do is add these probabilities together to get the answer and when we do we get the probability of getting exactly one head which is equal to 0.375 to check if this is a binomial experiment we have to see if it satisfies the for binomial conditions the first condition is satisfied because we have a fixed number of trials n is equal to 3 because the experiment has to be repeated 3 times in other words the coin has to be flipped three times the second condition is also satisfied because we can define a success as getting heads and we can define a failure as not getting heads this is the same as saying getting tails is a failure the third condition is also satisfied because the probability of success remained constant for every trial in other words the probability of getting heads is 0.5 and it stayed that way every time the coin was flipped and finally the fourth condition is also satisfied because each trial is independent gaining heads or tails for one trial doesn't change the probability of getting heads or tails for the other trials since all four conditions are satisfied we know that this is a binomial experiment now let's do a harder problem suppose we have 10 marbles in a box we have three pink marbles two green marbles and five blue marbles if we pick all five marbles with replacement what is the probability of drawing exactly two green marbles and is this a binomial experiment feel free to pause the video at this point so you can try this question for yourself before we calculate any probabilities let's see if we have a binomial setting is there a fixed number of trials yes n is equal to 5 because we are picking out 5 marbles are the two possible outcomes a success and a failure yes a success is getting a green marble and a failure is not getting a green marble is the probability of success constant for each trial yes because we are doing the experiment with replacement every time we conduct a trial or randomly pick out one marble we put it back into the box before drawing another marble if we didn't do this the trials would become dependent on one another instead of being independent in this case the probability of success is equal to the probability of getting one green marble which is equal to two over ten or 0.2 our trials independent of each other for the reasons mentioned previously the answer is yes since all four conditions are satisfied we know that this is the binomial experiment to solve this problem we need to write out all the possible ways of drawing exactly two green marbles for example one way is drawing a green marble on the first and second try and then not drawing a green marble on the third fourth and fifth try another way could be only drawing a green marble on the second and fifth try notice that I decided to write a dashed line for not getting a green marble this is because I don't care if the marble was blue or pink if I don't get a green marble it's a failure in a binomial setting we only care if we got a failure or a success overall we see that there are ten different ways of drawing two green marbles we can properly reframe this by saying that there are ten different ways of getting two successes and three failures now let's calculate some probabilities the probability of a success is equal to the probability of drawing a green marble since there are ten marbles in the box and only two of them are green the probability of drawing a green marble is equal to two over ten or 0.2 therefore the probability of a success is equal to 0.2 the probability of a failure is equal to their probability of not drawing a green marble there are 10 marbles in the box and eight of them aren't green so the probability of not drawing a green marble is just 8 over 10 or 0.8 therefore the probability of a failure is equal to 0.8 let's calculate the probability of the first outcome we have two successes followed by three failures so we will have 0.2 times 0.2 times 0.8 times 0.8 times 0.8 which gives us an answer of zero point zero two zero four eight if we do a similar calculation for the rest of the outcomes you should also get an answer of zero point zero two zero four eight notice how each of these ten outcomes has the same probability the reason for this is because each outcome has two successes and three failures so it makes sense that they all have the same probabilities now to calculate the probability of drawing exactly two green marbles all we have to do is add up all these probabilities together and when we do we get an answer of zero point two zero four eight however there is another way we can calculate this answer and that is by using the binomial formula the binomial formula looks like this it looks a little scary but let's break it down k is the number of successes n is the number of trials and little P is the probability of success what we have here in brackets represents the combination formula it is often seen in scientific calculators as NCR which is the same thing as saying n choose R so for the binomial formula we say that the probability of k is equal to n choose k times the probability of success little P raised to the number of successes K times the probability of failure which is 1 minus P raised to the number of failures which is equal to n minus K by using this formula we are essentially giving ourselves a nice shortcut for calculating things however it's important to remember that we can only use this formula if we have a binomial experiment let's go back to the problem so I can show you how to use this formula if we pickle five marbles with replacement what is the probability of drawing exactly two green marbles in this problem and is equal to five since there are five trials in other words we are randomly picking out five marbles K is equal to two since we are only concerned about getting exactly two successes this is the same as getting exactly two green marbles little P is equal to zero point two as this is the probability of success that we calculated earlier now all we have to do is plug these values into the formula and we'll get an answer so the probability of drawing exactly two green marbles in other words the probability of getting two successes is equal to 5 choose 2 times 0.2 squared times 1 minus 0.2 raised to the power of 5 minus 2 after this calculation we get an answer of zero point two zero four eight which is the same answer that we calculated from before if he found this video helpful consider supporting us on patreon to help us make more videos you can also visit our website at simple learning procom to get access to many study guides and practice questions thanks for watching

Channel: Simple Learning Pro

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Keywords: Statistics (Field Of Study), Statistics, stats, ap stats, ap statistics, Lesson, Tutorial, explained, math, mathematics, Simple Learning Pro, Practice Questions, binomial experiment, binomial setting, binomial formula, binomial, combination formula, formula

Id: nRuQAtajJYk

Length: 10min 6sec (606 seconds)

Published: Mon Jun 22 2020