 This is a video about criteria for using the binomial distribution. You need to know when it's appropriate to use the binomial distribution for calculating probabilities, and when it's not appropriate. There are four criteria which you need to remember, and these will tell you when it's appropriate to use the binomial distribution. The first criterion says that the number of trials must be fixed. The second one says that each trial must have the same two possible outcomes, a definite success or a definite failure. Thirdly, the trials must be independent, so the outcome of one trial doesn't affect the probability of success in the next trial. And more generally, the probability of success must be the same in every single trial. Okay, I want to look at some examples to help you understand these criteria. My first example has to do with tossing a coin. Suppose that we toss a coin ten times, and we're interested in the number of heads. Let's go through the criteria and see whether they apply. Well, first of all, the number of trials is definitely fixed, because we said that we would toss the coin ten times. Secondly, it's true that each trial has the same two possible outcomes, because each time we either get a head or a tail. Thirdly, the trials are independent, because if I get a head one time, that doesn't make any difference to the probability of getting a head the next time. And finally, the probability of success is the same in each trial, because each time the probability of getting a head is the same. So in this case, all four criteria apply, and therefore it's appropriate to use the binomial distribution. My next example has to do with cards. Suppose that we draw ten cards at random from a pack, but don't replace them. And then we say that x is the number of kings. Let's go through the criteria again. Well, the number of trials is fixed, because I said we would take ten cards. And each trial does have the same two possible outcomes, because each time we can either get a king or some other card. But the trials are not independent, because the cards are drawn without replacement. And therefore the probability of getting a king changes depending on what cards are drawn. For example, if the first four cards drawn are all kings, then the probability of the fifth card being a king is obviously zero. On the other hand, if none of the first four cards drawn are kings, then the probability that the fifth card is a king will be four out of 48. The fact that the probability of getting a king changes according to what cards have already been drawn means that the trials are not independent. It is true that the probability of success is the same in each trial, because if we don't know anything about the outcome of previous trials, then the probability of getting a king is always one in thirteen. But the fact that one of these criteria doesn't apply means that we can't use the binomial distribution to calculate probabilities in this example. For my next example, think of Serena Williams playing a tennis match. Suppose we're interested in the number of games that she wins. This time, it's not even true that the number of trials is fixed, because the total number of games that are played varies from match to match. It is true that each trial has the same two possible outcomes, because each game is either one or lost. But it's probably not true that the trials are independent. I suspect that if a player wins several games in a row, then their confidence increases, and they're more likely to win the next game. It's certainly not true that the probability of success is the same in each trial. Firstly, because the probability of winning is much higher when you're serving than when your opponent is serving. And secondly, because players seem to go through phases in a match, whether doing better or worse. So the probability of success is higher when you're serving, and also when you're playing a bit better. My next example is to do with children, and in particular whether they're boys or girls. Let's suppose we're interested in the number of girls in a family. We'll go through each criterion and see whether it applies. The first one doesn't. The number of trials isn't fixed, because different families have different numbers of children. It is true that each trial has the same two possible outcomes, because each child is either a girl or not. The third criterion is interesting. Are the trials independent? In other words, does the probability that a child is a girl depend on whether the previously born siblings were boys or girls? Well, there's one clear reason why the trials can't be independent, and that's to do with identical twins. In some cases, a child will be genetically identical to a previously born child, and in that case, it's bound to be the same sex. But even apart from the issue of identical twins, it seems that the trials are not independent. The probability of having a girl depends on the sex of previously born children. For example, if the child is the first to be born, then the probability that it's a girl is 49%. Whereas if the child is the fourth to be born and the previous children were all boys, then the probability that the fourth child is a girl is only 43.6%. On the other hand, if the child is the third to be born and the previous two children were both girls, then the probability that the third child is a girl is 54%. This shows that the trials are not independent. It's also the case that the probability of success is not the same in each trial. You can see that if it's the first child, then the probability of having a girl is 49%. Whereas if it's the second child, it's only 48.8%. If it's the third child, it's 51.4%. And if it's the fourth child, then the probability is 49.2%. So the probability of success is not the same in each trial. This shows that the number of children in a family that are girls does not have the binomial distribution. And you can't work out probabilities in this situation by using the binomial distribution. My last example is to do with politics. Suppose we choose 100 adults at random, and we count the number who say that they would vote conservative. In this situation, the number of trials is fixed because we're asking 100 people. It's also true that the trials have the same two possible outcomes. Either they say that they will vote conservative, or they say something else. Note that we do have to be careful what we count as success here, though. For example, if we were trying to count the number of people who will actually vote conservative in the next election, then we'd have a problem, because some of them might refuse to say, and then we wouldn't have a definite success or a definite failure. The trials are independent because the probability of one person saying that they will vote conservative is not affected by somebody else saying that. Assuming that we pick these people randomly and that they don't talk to each other or know what one another have said. Finally, the probability of success is the same in each trial as long as we pick the people randomly. So this is a situation where the binomial distribution does apply, and we can use it to work out probabilities. So there are four criteria that you need to remember so that you know whether or not the binomial distribution applies. These are that the number of trials must be fixed. Each trial must have the same two possible outcomes, a definite success or a definite failure. The trials have to be independent, and the probability of success must be the same in each trial. This is the end of my video about criteria for using the binomial distribution. Thank you for watching.