Editing Fisher's exact test (section)

=== Second example ===
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 <math>\mathfrak{p}</math>, the probability that a woman is a studier is also <math>\mathfrak{p}</math>, 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 conditional probability of observing the values ''a, b, c, d'' in the four cells, conditionally on the observed marginals (i.e., assuming the row and column totals shown in the margins of the table are given). This remains true even if men enter our sample with different probabilities than women. The requirement is merely that the two classification characteristics—gender, and studier (or not)—are not associated.

For example, suppose we knew probabilities <math>P, Q, \mathfrak{p,q}</math> with <math>P + Q = \mathfrak{p} + \mathfrak{q} = 1</math> such that (male studier, male non-studier, female studier, female non-studier) had respective probabilities <math>(P\mathfrak{p}, P\mathfrak{q}, Q\mathfrak{p}, Q\mathfrak{q})</math> for each individual encountered under our sampling procedure. Then still, were we to calculate the distribution of cell entries conditional given marginals, we would obtain the above formula in which neither <math>\mathfrak{p}</math> nor <math>P</math> occurs. Thus, we can calculate the exact probability of any arrangement of the 24 teenagers into the four cells of the table, but Fisher showed that to generate a significance level, we need consider only the cases where the marginal totals are the same as in the observed table, and among those, only the cases where the arrangement is as extreme as the observed arrangement, or more so. ([[Barnard's test]] relaxes this constraint on one set of the marginal totals.) In the example, there are 11 such cases. Of these only one is more extreme in the same direction as our data; it looks like this:

{|class="wikitable"  style="text-align:center;"
|-
!
! &nbsp;&nbsp; Men &nbsp;&nbsp;
! &nbsp; Women &nbsp;
|''Row Total''
|-
!scope="row" | Studying
|bgcolor="lightgray" | '''0''' ||bgcolor="lightgray" | '''10''' || ''10''
|-
!scope="row"| &nbsp; Non-studying &nbsp;
|bgcolor="lightgray" | '''12''' ||bgcolor="lightgray" | '''2''' || ''14''
|-
| ''Column Total''
| ''12'' || ''12'' ||  ''24''
|}
For this table (with extremely unequal studying proportions) the probability is
<math>{p =  {\tbinom{10}{0}} {\tbinom{14}{12}} }/{ {\tbinom{24}{12}} } \approx 0.000033652</math>.