a b a + b c d c + d a + c b + d N 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 (N_t) 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 (N_c) 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 (N_t) by the number of ways of obtaining the cell frequencies you found (N_c).

P = N_t / N_c = {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.