Stats: Binomial Probability Distribution (Part 1)

this video goes through and discusses the high level the overall picture of what we mean by a binomial probability distribution now the first thing that I want to point out to you is this prefix by right here that prefix bi which simply means and that's no surprise to you simply means the number two right bicycle or bifocals or anything with a bi so what we mean when you think when you hear that phrase binomial probability distribution I want you to think of two possible outcomes alright two possible outcomes something called a success and the other outcome is a failure the other thing with binomial probability distributions that come to mind is and so let me put here two possible outcomes okay we'll call them success refer a little bit more to this in a second and failure the other thing I want you to think of is that binomial probability distributions have a fixed number of trials right as a fixed number of trials for example let's use an analogy of say basketball like basketball so when I stand at the free-throw line let's say I was I was I was going to make a shot I was going to shoot the basket but I got fouled you know that in basketball you get if the foul was called by the referee you get two free shots at the free-throw line right so that would be the number two would be the fixed number of outcomes or the fixed number rather of trials if I was standing behind the 3-point line and trying to shoot and got fouled I would have a fixed number of trials of three I'd be able to take three shots from the free-throw line okay anyway so the there is going to be a set number of trials and I'm going to talk about that in just a second but the the set number of trials is represented by the lowercase n number oops number of trials okay the other thing to that that were of interest is X now what's X going to represent X is going to represent the number of times right so given n trials the number of times that I succeed that I actually have some success okay so X can be any number X is going to represent the number of successes and that's going to be any number between 0 and n I hope you understand okay so if I'm standing at the free-throw line and I get three shots I get two three shoot three times and I want to let's say I want to make at least two out of those three or something like that or looking for the probability that I make exactly just one or you know X is going to be how many shots I get to make right how many shots actually go in the hoop okay so X can be zero maybe I don't make any shots or it could be one or it could be two or it could be three I cannot I cannot make four successes alright I cannot make four shots if I'm only taking three at most all right so and it's going to be the number of trials and X is going to represent the number of times I succeed for each of these trials all right the other thing is that when I stand at the free-throw line and I make a basket there is something called a little p and a little Q some books don't use Q some authors just use a P and I'm going to talk about P and Q in just a second but little P is going to stand for the probability of success one of my favorite basketball players is Ray Allen for the Boston Celtics and he is a great free-throw shooter he shoots hundreds of free-throws almost every single day and so he's really really good every time he stands at the line he stands probably even greater than this but let's just make it a nice even number he stands a 90% chance of making a shot from the free-throw line so that's his probability of success well what is Q q is the complement of P all right Q and P are complements of each other so if you know the probability of success is 0.9 then the complement of it these guys are complements of each other the complement of it then is 1 minus that we scroll it down so you can see my word complement the complement is 1 minus P right 1 minus little P or in this case 1 minus point 9 is point 10 so he stands a 10% chance of missing and a 90% chance of making it okay so pulling all of these things together there is a formula that can tell you what the probability is of making X number of successes out of n independent trials now this formula is a little bit daunting don't be don't be too alarmed or freaked out when you see this formula it's really not that bad especially if you've got some rudimentary statistics or probability under your belt but the formula goes like this all right the probability of making X successes is equal to n factorial over N minus X factorial times X factorial that's a combination formula right there if you recognize that times the probability of success raised to the X power times the probability of failure raised to the N minus X power all of all three of these things this one this one and this one get multiplied together and that will give you the probability of X successes out of n trials now that formula is amazingly nasty all right it's actually not that bad especially if you have a calculator the good thing is is that some books supply a table for you that can help you find out and it does all this number crunching for you now there are so many variables in there that a table can't be big enough to house every single one of them but take a look at this table I'm using the Triola book this is the 11th edition and in the back of the book table a1 is a binomial probability table and I know it's hard to see but that number right there is 13 all right that's an N number anything in this first column is an n number that's a number 14 and all of this second column here represents an X so where I'm pointing that's an X number and it goes from 0 all the way up to 13 okay see how that works and again like I said that the number of successes has to be between 0 and n right I can't make I can't make 14 free-throws if I'm only taking 13 free-throws that just doesn't make sense so that's why this table in the X column goes between 0 and whatever your n value is my n value is 13 that's what's showing up here is 13 at the very bottom okay now what are all these numbers over here these numbers correspond to little P which is the probability of success right where that little p is the probability of success so let me work out an example here let's try something o say like this let's say that I am looking for all right let's say that I'm looking for the probability that X is equal to 8 all right the probability that X is equal to 8 given right and you can't see that there we go X is equal to 8 given that n is equal to 13 so I'm going to repeat this trial 13 times and that the probability of success is 0.5 as soon as you see that point 5 by the way you automatically know that Q is also 0.5 right because these two things are complements of each other ok so if I'm looking for the probability that X is equal to 8 I could plug it into that formula that I just showed you or I could go and - this table right here or I can refer to this table I know you can't read those numbers because they're awfully blurry and I'm sorry I can't focus in on that but I'm just going to point something out to you probably have a table similar to this in your book all right what I'm going to do is this n is 13 all right this n is 13 that's the number of trials that I have I'm looking for probably that X is equal to 8 there's my X is equal to 8 and the probability of success was 0.5 so I'm just scooting over here and I'm looking and I can see that this is 0.15 7 at that number that I'm pointing to with my fingers 0.15 7 so using this information that I was given I know that the probability of this thing is equal to 0.15 7

Channel: poysermath

Views: 545,354

Keywords: binomial, probability, distribution, exact, success, trials, poysermath

Id: gAkyLGuUbh0

Length: 9min 39sec (579 seconds)

Published: Tue Mar 27 2012