Given a total of N items K of which are type1 items the prob

Given a total of N items, K of which are type-1 items, the probability to observe a subset of k type-1 items in a subset of n items selected from the total N items without repetitions is given by the hypergeometric probability distribution

The corresponding p-value is defined as the probability to observe the subset of k or more than k type-1 items in a subset of n items selected from the total N items without repetitions. Use combinatorial proofs to show that the three formulations below are all mathematically equivalent formulations of the p-value:

Solution

a) M(say) gives the sum of all Ps from k to K.

If S represents the probability from 1 to k-1 from the theorem of total prob =1

we get M+S =1

Or M = 1-S

1-S is nothing but the terms represented by b.

Hence proved that a and b are equivalent.

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To prove that b = c

i can take values from 0 to N and K can also take values from 0 to N i.e.

from 0 type I items to all N type II items

K = N if all are of type I items

Hence replace in a K by N to get c

Thus all 3 are equivalent.