hey guys it's mark from ace tutors and in this video we are going to continue our discussion on different probability distributions this time i'll be going over the binomial distribution so first let's go over what makes an experiment a binomial distribution in order to be a distribution of this type it must satisfy each letter of this acronym bins the first letter b stands for binary outcomes and what this means is that there are only two outcomes of your given event either a certain outcome occurs usually called a success or it does not occur usually called a failure next the binomial distribution must have independent trials in other words the successor failure of one event must not somehow affect the success or failure of any other event then this distribution must also have a defined n number of trials and finally each trial must have the same probability of success each time so for example let's say you were rolling a die four times and you wanted to find the probability of rolling a one two of those four times let's see if this experiment meets the requirements for a binomial distribution so first during each row you can put the outcome into one of two categories either you roll a one aka a success or you don't roll a one aka a failure so it satisfies the b next the fact that you roll a certain number during one trial has no effect on the number you roll on the next trial so this also satisfies the eye then we are rolling the die four times so we specifically have four trials and that satisfies the n and finally during each trial we have the same probability of one-sixth for rolling a one so that satisfies the s and because all these conditions were met we know this experiment will follow a binomial distribution now that we covered what makes something a binomial distribution let's go over some properties of this distribution first the mean or expected value for the binomial distribution the mean mu is pretty simple and equals n times p where n is the number of trials for the experiment and p is the probability of success for each trial next the standard deviation is also pretty straightforward sigma equals the square root of n times p times 1 minus p or you might also see it as the square root of n times p times q where q equals one minus p aka the probability of failure since there are only two outcomes for a trial finally let's go over some of the more interesting stuff probability if you want to calculate the probability of getting x number of successes in your experiment all you do is n choose x times p raised to x times q raised to n minus x uh what to see what the heck this means and where it comes from let's dissect it a bit with our previous example of rolling a die 4 times as a reminder in this experiment we want to find the probability of rolling a 1 2 out of our 4 trials so first let's look at what this right side of the formula gives us with our n equaling 4 trials our probability of success equaling 1 6 and our desired number of successes equaling 2 we can write this as 1 6 squared times 5 6 raised to 4 minus 2 or 5 6 squared we were to expand this out it would be 1 6 times 1 6 times 5 6 times 5 6. but what does this mean well since each of our four trials are independent to find the probability of a certain situation occurring we can multiply together the probabilities of the outcomes for each trial so in this order we would specifically be finding the probability of getting a success aka rolling a one on the first roll a success on the second roll a failure on the third roll and then finally a failure on the fourth roll so essentially this right side is a way to calculate the probability of one specific situation that satisfies our desired outcome of rolling a one twice but there are other situations that satisfy this as well right what if we roll the one on our first and third rolls or on our third and fourth rolls or any of the other situations that result in two ones how do we account for all those possibilities well that's where this thing comes into play which is pronounced as n choose x basically what this value gives us is the total number of different ways you can write your desired outcome in other words out of our n trials how many ways are there to have x successes well for our example out of four roles how many ways are there to roll a 1 twice so to calculate and choose x you need to look at one more ugly looking formula but just like the other one it's not as scary as it looks if you haven't seen them before these exclamation points represent factorials and all the factorial is is taking a number and multiplying it by each integer below it until you reach one so the factorial of three would be three times two times one or equal to 6. and if we wanted to calculate 4 choose 2 for our example we would take 4 factorial and divide it by 4 minus 2 factorial times 2 factorial as a result our top would be four times three times two times one and our bottom would be two times one times two times one and this equals 24 over four or six so this means that out of our four roles there are six different ways to roll a one twice overall in order to find the probability of a certain outcome we take the probability of reaching that outcome in one specific way and multiply it by the total number of possible ways there are to write that desired outcome and to see in its full beauty the probability of rolling two ones is equal to four choose two times one-sixth squared times five-sixths raised to four minus two using our previous calculations this simplifies to six times 1 6 squared times 5 6 squared or 0.1157 so the probability of rolling one twice in four rolls is 11.57 percent and just for fun let's also find the mean and standard deviation of this scenario once again to find the mean we multiply n times p or for our example 4 times 1 6 to get that mu equals 4 6 and what this means is that during your 4 rolls you can expect to roll a 1 4 6 times and this might seem a little strange because you can't roll a 1 4 6 of a time but if you were to repeat this experiment of 4 rolls again and again and average the number of times you rolled a 1 per experiment the value you get would approach 4 6. next to calculate the standard deviation we would take the square root of n times p times 1 minus p which would be the square root of 4 times 1 6 times 1 minus 1 6 or 4 times 1 6 times 5 6 or equal to 0.745 now i hope you found this video helpful in your journey to master statistics but if you have any questions or feedback for how we can make our videos better please let us know down in the comments and if you did find this helpful please tap those like and subscribe buttons to see more of our content thanks again for watching and remember you have big dreams don't let a class get in the way