An urn contains t1 balls of color C1 t2 of color C2 tk o

An urn contains t1 balls of color C1, t2 of color C2, . . . , tk of color Ck. If n balls are drawn without replacement, find the probability of obtaining exactly n1 of color C1, n2 of color C2, . . . , nk of color Ck. (Note: the resulting expression, regarded as a function of n1, n2, . . . , nk, is called the hypergeometric probability function.)

Solution

Here given that urn contains t1 balls of color C1, t2 balls of color C2 ,.......,tk balls of color Ck.

and we have to find the probability of obtaining exactly n1 color of C1,n2 color of C2.........nk color of Ck.

Here N=t1+t2+.........tk

and n=n1+n2+.......+nk

First we find exactly n1 balls of color C1 from t1 because there are total t1 balls of color C1.

So in notation we write it as (t1 C n1) =>It means we select n1 balls from t1.

Similarly we find n2 balls of color C2 from t2=>

In notation (t2 C n2)=>n2 balls fom t2

Similarly we find for nk balls of color Ck=>

In notation (tk C nk)=>nk balls from tk.(Note : C is used for combination)

Different number of balls shows different color which is drawn from it.

That means t1,t2,.................,tk all are independent.

P(exactly n1 color of C1,n2 color of C2,........,nk color of Ck) = (t1 C n1)*(t2 C n2)*...........*(tk C nk) /(N C n)

This is an answer.