A bin has 8 black balls and 7 white balls.  3 of the balls are drawn at random.  What is the probability of drawing 2 of one color and 1 of the other color?
The number of ways to draw out 3 balls from 15 is $\binom{15}{3}=455$.  We can choose 2 black balls and 1 white ball in $\binom{8}{2}\binom{7}{1}=196$ ways.  We can pick 1 black ball and 2 white balls in $\binom{8}{1}\binom{7}{2}=168$ ways.  Therefore we have $196+168=364$ ways to satisfy the condition, so the answer is $\dfrac{364}{455}=\boxed{\frac{4}{5}}$.
\frac{4}{5}
Level 5
Counting & Probability
Problem: A bin has 8 black balls and 7 white balls.  3 of the balls are drawn at random.  What is the probability of drawing 2 of one color and 1 of the other color?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability of drawing 2 balls of one color and 1 ball of another color, we first identify the total number of balls, which is 15 (8 black + 7 white). The total number of ways to draw 3 balls from 15 is calculated using the combination formula C(n, k), where n is the total number of items, and k is the number of items to choose. Thus, C(15, 3) = 455.

Next, we consider the two scenarios for drawing 2 balls of one color and 1 ball of another color. The first scenario is drawing 2 black balls and 1 white ball. The number of ways to choose 2 black balls from 8 is C(8, 2) = 28, and the number of ways to choose 1 white ball from 7 is C(7, 1) = 7. Therefore, the total ways for this scenario is 28 * 7 = 196.

The second scenario is drawing 2 white balls and 1 black ball. The number of ways to choose 2 white balls from 7 is C(7, 2) = 21, and the number of ways to choose 1 black ball from 8 is C(8, 1) = 8. Thus, the total ways for this scenario is 21 * 8 = 168.

Now, we sum the total ways from both scenarios: 196 (for 2 black and 1 white) + 168 (for 2 white and 1 black) = 364.

Finally, to find the probability of drawing 2 balls of one color and 1 of the other color, we divide the total favorable outcomes (364) by the total outcomes (455). Therefore, the probability is 364 / 455. Simplifying this fraction gives us 4 / 5.<|END_OF_LATENT|> The final answer is $\frac{4}{5}$.