Question

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?

Answer 1

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.