 In a binomial experiment, we repeat a simple experiment a fixed number of times, an outcome of interest has a fixed probability, and we're interested in the number of times an outcome of interest occurs. But sometimes we repeat an experiment until an outcome of interest occurs a number of times. For example, calling customers until you make five sales, drawing cards until you get a pair, litigating until you get your way. And this gives us a negative binomial probability. So for example, suppose you flip a fair coin until you get three heads, find the probability this will take five flips. So to understand the negative binomial distribution, here's an obligatory riddle. Why is it that your keys are always in the last place you look for them? And the answer is that once you find them, you stop looking. And this is an important idea to keep in mind when looking at these negative binomial probabilities. If it took us five flips to get three heads, then we know that the fifth flip had to be the third head. And in fact, we can go a little bit further. In order for it to take five flips to get three heads, the fifth flip had to be a head result, and the first four flips had to have two heads. And this is the only way it would have taken us five flips to get three heads. If the first four flips had more than two heads, we would have stopped earlier. And if the fifth flip wasn't a head, then we'd keep going. And so what we sometimes call the negative binomial probability really is a special case of a binomial probability. So let's analyze that. The probability of getting exactly two heads in four flips is a binomial probability, and we can calculate that to be. And the probability of getting a head result on the last flip is 0.5. And so the probability that it takes five flips to get three heads is the probability that we got exactly two heads on the first four flips, 0.375, times the probability we got a third head on that last flip, 0.5, and our probability is... Now we can derive a formula for the negative binomial probability, but it's actually easier to reason your way through it, and there's less chance of making a mistake somewhere along the way. For example, suppose a con artist could get a mark to give them their vote. I mean, suppose they can con a mark to give them their money with probability 0.05. What is the probability the con artist will have to talk to at least ten people before finding two victims? So again, if the keys are in the last place they look, if they need to talk to at least ten people, well, let's rewrite that, ten or more people to find two victims. This means they didn't find two victims among the first nine. In other words, of nine people, zero or one were swindled. So the probability that zero of the first nine were swindled, again, that's a binomial probability, and we can calculate that. The probability that one of the first nine were swindled, again, a binomial probability that we can calculate, and this event, ten or more people, is the same as saying of nine people, zero or one were swindled. And so those probabilities give us about a 93% chance that this con artist will need to talk to at least ten people.