A bag contains 20 white marbles and 30 black marbles. if 7 marbles are chosen, what is the probability that there will be 2 white marbles and 5 black marbles?

This is a hypergeometric distribution problem.

Population (N=50=W+B) is divided into two classes, W (W=20) and B (B=30).

We calculate the probability of choosing w (w=2) white and b (b=5) black marbles.

Hypergeometric probability gives

P (W, B, w, b) = C (W, w) C (B, b) / (C (W+B, w+b)

where

C (n, r) = n! / (r! (n-r) !) the number of combinations of choosing r out of n objects.

Here

P (20,30,2,5)

=C (20,2) C (30,5) / (20+30, 2+5)

=190*142506/99884400

=0.2710

Alternatively, doing the combinatorics way:

#of ways to choose 2 from 20 = C (20,2)

#of ways to choose 5 from 30=C (30,5)

total #of ways = C (50,7)

P (20,30,2,5) = C (20,2) * C (30,5) / C (50,7)

=0.2710

as before.