Editing Fisher's exact test (section)

=== First example ===
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:

{|class="wikitable" style="text-align:center;"
|-
!
! &nbsp;&nbsp; Men &nbsp;&nbsp;
! &nbsp; Women &nbsp;
|''Row Total''
|-
!scope="row" | Studying
|bgcolor="lightgray" | '''''a''''' ||bgcolor="lightgray" | '''''b''''' || ''a + b''
|-
!scope="row"| &nbsp; Non-studying &nbsp;
|bgcolor="lightgray" | '''''c''''' ||bgcolor="lightgray" | '''''d''''' || ''c + d''
|-
| ''Column Total''
| ''a + c'' || ''b + d'' ||  ''a + b + c + d (=n)''
|}

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:

<div class="center">
<math>p = \frac{ \displaystyle{{a+b}\choose{a}} \displaystyle{{c+d}\choose{c}} }{ \displaystyle{{n}\choose{a+c}} } = \frac{ \displaystyle{{a+b}\choose{b}} \displaystyle{{c+d}\choose{d}} }{ \displaystyle{{n}\choose{b+d}} } = \frac{(a+b)!~(c+d)!~(a+c)!~(b+d)!}{a!~~b!~~c!~~d!~~n!}</math>
</div>

where <math> \tbinom nk </math> is the [[binomial coefficient]] and the symbol ! indicates the [[factorial|factorial operator]].
This can be seen as follows. If the marginal totals (i.e. <math>a+b</math>, <math>c+d</math>, <math>a+c</math>, and <math>b+d</math>) are known, only a single degree of freedom is left: the value e.g. of <math>a</math> suffices to deduce the other values. 
Now, <math>p=p(a)</math> is the probability that <math>a</math> elements are positive in a random selection (without replacement) of <math>a+c</math> elements from a larger set containing <math>n</math> elements in total out of which <math>a+b</math> are positive, which is precisely the definition of the hypergeometric distribution.

With the data above (using the first of the equivalent forms), this gives:

<div class="center">
<math>p = { {\tbinom{10}{1}} {\tbinom{14}{11}} }/{ {\tbinom{24}{12}} } = \tfrac{10!~14!~12!~12!}{1!~9!~11!~3!~24!} \approx 0.001346076</math>
</div>