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Those margin totals


aba + b
cdc + d
a + cb + dN

Fisher's formula for the probability of a given combination of cell frequencies (a,b,c,d as shown) in a 2 2 contingency table is derived from two terms:

  1. For any specified set of margin totals, the total number of ways to achieve them (Nt) is given by:
    Nt =                N!²            
    (a+b)! (c+d)! (a+c)! (b+d)!
  2. The number of ways of achieving a specified cell combination (Nc) is given by:
    Nc =          N!      
    a! b! c! d!

Therefore, to find what proportion (P) of occasions a given combination of cell frequencies can be expected to arise, you divide the number of ways you can achieve the margin totals (Nt) by the number of ways of obtaining the cell frequencies you found (Nc).

P = Nt / Nc = {N!² / [(a+b)! (c+d)! a+c)! (b+d)!]} / {N! / [a! b! c! d!]}

Rearranging this expression we get
P = {N! (a+b)! (c+d)! (a+c)! (b+d)!} / {N!² a! b! c! d!}

Cancelling the extra N! gives us
P =    (a+b)! (c+d)! (a+c)! (b+d)!
N! a! b! c! d!

This is Fisher's standard formula.