in the last set of videos we talked about discrete probability distributions in this set of videos we're going to talk about a particular kind of discrete probability distribution called a binomial distribution to be able to describe a binomial distribution it's helpful to first know what kind of experiment the probability distribution describes what outcomes of what experiment this binomial distribution describes okay um and so we want to first talk about a binomial experiment now there are three things that are crucial for running a binomial experiment one you run a certain fixed number of Trials there are a fixed number of repeated trials and we usually notate this with the letter N the number of trials in so for example say we roll the dice every time we roll the dice is like a trial okay a standard six sided dice if we roll the dice 10 times then there would be 10 trials so if we roll a six-sided dice 10 times then there are 10 trials okay two the second requirement is that in each trial there's only two possibilities okay each trial has only two possible outcomes and we call these outcomes success and failure we notate the probability of success with the letter p the probability of success is p now because there's only two possible outcomes that means the probability of failure is the complement of success right you could only do the two things so together if we look at the probability of success and probability of failure and look at them together it must be 100 of all the outcomes of this particular trial right so the probability of failure is 1 minus p because 100 represents all you know everything the chance that anything happens at all right there's a 100 chance that something happens here and all we're doing is we're taking away the probability of success what's left over must be the probability of failure since those are the only two possibilities okay um let's go back to our dice rolling example say we are interested in rolling a 2. we are interested in whether or not the dice lands on two for each trial okay so then the probability that the dice lines went to the probability of success is one out of six because there's one way that the dice can land on two out of six possible sides right and the probability of failure which in this case would be the probability that the dice lands on anything other than two would be 1 minus 1 6 or 5 6. so that's the probability of success and the probability of failure notice we get to kind of determine we Define ourselves what success means and what failure means success doesn't necessarily have to be good failure doesn't have to be bad success just means what are we trying to look for what are we trying to measure okay if that outcome happens that's a success so it could actually be quite a bad thing if um if we're measuring the number of instances that people get cancer we might call a success someone getting cancer that's not a good thing right but uh that's what we are measuring for okay so here there are obviously lots of other numbers we could have landed on it other than two but in this case that's all we're looking for so 2 is all we're looking for if that happens great if not then we count it as a failure so everything else is considered anything other than two is considered the same kind of outcome that's a failure okay and then one more requirement of a binomial experiment okay the trials the end trials of a binomial experiment are independent and identical pendant and identical okay so every time we run a trial it's like uh it's the same thing that's happening so if we're rolling that dice 10 times okay if we roll a dice n equals 10 times each roll is independent and identical to every other Rule and dependent and identical to every other role there's not like s a special role right every time we roll the dice it's the same thing that's happening right there's a one out of six chance that we'll get a two there's a five out of six chance it will roll something that's not a two which was why this would be a binomial experiment so a binomial probability distribution a binomial probability distribution is a formula that tells the outcome tells the probability that the outcome of a given binomial experiment results in X successes out of n trials let's go to binomial probability distribution is if we roll that die 10 times the binomial probability distribution will be able to tell us the probability uh that out of those 10 rules that three of them will be twos or that nine of them will be twos or that five of the ten rules will be twos okay that's what the binomial probability distribution does for us now we'll talk about how we actually go about calculating these probabilities in a future video for now I want to focus on the expected value and the standard deviation of a binomial probability distribution we learned in the last set of videos how to calculate the expected value of the mean and the standard deviation of a discrete probability distribution this is a particular kind of discrete probability distribution but it turns out for this particular distribution we can simplify the formula considerably okay so the expected value for a binomial distribution is just n times P where again n is the number of Trials and P is the probability of success if you think about it this kind of makes sense right if you have six trials and your probability of success is one out of six then you would expect one of those roles to end up being whatever you're interested in whether it's a two or what not okay so the probability is the long-term proportion so this is just saying how many if we if we ran this experiment lots of times however Big N is if we repeat those trials lots of times how many times would we expect this result to happen Okay how many times would we expect a success and then the standard deviation is a little bit different it's the square root of n times p times 1 minus p so that's the square root of the number of Trials times the probability of success of any one trial times the probability of failure of any one trial