Problem Solving

There are 15 slate rocks, 20 pumice rocks, and 10 granite rocks randomly distributed in a certain field. If 2 rocks are to be chosen at random and without replacement, what is the probability that both rocks will be slate rocks?

A) 1/3
B) 7/22
C) 1/9
D) 7/66
E) 2/45

Answer should be D.

This is a fairly basic one, anyway here is the solution

We Know,
Probability for the Event n(P) = n(E)/n(S)

Where n(E) = Number of events that the two rocks selected are Slate stone
n(S) = Sample Size ( number of ways to pick any 2 stones)

Here,
n(E) = 15C2 = 15!/(2!*13!) = 15*7
n(S) = 45C2 = 45!/(2!*43!) = 45*22
Therefore,
n(P) = n(E)/n(S) = (15*7)/( 45*22) = 7/66 ………………. Ans

Another way of thinking about is:

1. Probability of getting first slate rock: 15/45 = 1/3
2. Probability of getting 2nd slate rock: 14/44 = 7/22

Thus: (1/3)*(7/22) = 7/66