# Fishers test - true probability

We suppose $a_{11} = min \{ a_{kl}; k \in 1..numberOfRows, l \in 1.. numberOfColumns\}$

$P = \sum_{i=0}^{a_{11}} \frac{r_1! r_2! s_1! s_2!} { n! (a_{11}-i)! (a_{12}+1)! (a_{21}+i)! (a_{22}-i)!}$

Questions is why P represents PROBABILITY that random variables A,B are independent?

We usually reject hypothesis that that A,B are independent if the P is smaller $\alpha = 0.05$

Example

• Original table $a_{11}$ represents that number of people injured on place A.

 Injured Survived A 5 10 15 B 2 38 40 7 48 55

• Switched rows to get minimum at $a_{11}$:

 2 38 40 5 10 15 7 48 55

• P0:

 0 40 40 7 8 15 7 48 55

• P1:

 1 39 40 6 9 15 7 48 55

• P2:

 2 38 40 5 10 15 7 48 55

• Overall probability: $P = P_0 + P_1 + P_2$