One black, one white, one gray marble

Question

A bag holds 3 black marbles, 3 white marbles, and 4 gray marbles, 10 in total. You draw 3 marbles at random. What is the probability of drawing exactly one black, one white, and one gray?

Accepted Answer

Choose one from each color's pile, multiply, then divide by all ways to draw 3. Here \binom{n}{k} counts ways to choose k from n. Total ways to draw 3 from 10: \binom{10}{3}=120. Favorable ways: \binom{3}{1}	imes\binom{3}{1}	imes\binom{4}{1}=3	imes 3	imes 4=36. Divide: \frac{36}{120}=\frac{3}{10}=0.3. Takeaway. Three draws hitting three different colors is often surprisingly likely when the piles are close in size. Multiply one count per color and divide by the total ways to draw.

One black, one white, one gray marble

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