what is the probability that she will select a blue marble first, then a clear marble?

Question

Maria has eight black marbles, fourteen clear marbles, and twelve blue marbles in a bag. If she picks two marbles at random, without replacement, what is the probability that she will select a blue marble first, then a clear marble?

Answer 1

This is a compound probability without replacement, which means that the first event removes an item at least from the denominator, and in some cases, from the numerator as well. There are 34 total marbles and 12 blue ones, so the probability of the first event is ¹²/₃₄ = 6/₁₇. The second event is drawing a clear marble from a bag that has one less blue marble. This makes the grand total number 33, but the number of clear marbles is unaffected. This probability will be ¹⁴/₃₃. So the probability of the two events together is

6/₁₇ • ¹⁴/₃₃ = ²⁸/₁₈₇ ≈ 0.15.