than among the men, and we want to test whether any difference in proportions that we observe is significant. The data might look like this: The question we ask about these data is: Knowing that 10 of these 24 teenagers are studying and that 12 of the 24 are female, and assuming the null hypothesis that men and women are equally likely to study, what is the probability that these 10 teenagers who are studying would be so unevenly distributed between the women and the men? If we were to choose 10 of the teenagers at random, what is the probability that 9 or more of them would be among the 12 women and only 1 or fewer from among the 12 men? Before we proceed with the Fisher test, we first introduce some notations. We represent the cells by the letters a, b, c and d, call the totals across rows and columns marginal totals, and represent the grand total by n. So the table now looks like this: Fisher showed that conditional on the margins of the table, a is distributed as a hypergeometric distribution with a+c draws from a population with a+b successes and c+d failures. The probability of obtaining such set of values is given by: where is the binomial coefficient and the symbol ! indicates the factorial operator. This can be seen as follows. If the marginal totals (i.e. , , , and ) are known, only a single degree of freedom is left: the value e.g. of suffices to deduce the other values. Now, is the probability that elements are positive in a random selection (without replacement) of elements from a larger set containing elements in total out of which are positive, which is precisely the definition of the hypergeometric distribution. With the data above (using the first of the equivalent forms), this gives: The formula above gives the exact hypergeometric probability of observing this particular arrangement of the data, assuming the given marginal totals, on the null hypothesis that men and women are equally likely to be studiers. To put it another way, if we assume that the probability that a man is a studier is , the probability that a woman is a studier is also , and we assume that both men and women enter our sample independently of whether or not they are studiers, then this hypergeometric formula gives the