A bag contains 10 red marbles and 6 blue marbles. Three marbles are selected at random and without replacement. What is the probability that one marble is red and two are blue? Express your answer as a common fraction.
There are three ways to draw two blue marbles and a red one: RBB, BRB, and  BBR. Since there are no overlapping outcomes, these are distinct cases and their sum is the total probability that two of the three drawn will be blue.  The desired probability therefore is  \[\frac{10}{16}\cdot\frac{6}{15}\cdot\frac{5}{14} + \frac{6}{16}\cdot\frac{10}{15}\cdot\frac{5}{14} + \frac{6}{16}\cdot\frac{5}{15}\cdot\frac{10}{14} = \boxed{\frac{15}{56}}.\]
\frac{15}{56}
Level 4
Counting & Probability
Problem: A bag contains 10 red marbles and 6 blue marbles. Three marbles are selected at random and without replacement. What is the probability that one marble is red and two are blue? Express your answer as a common fraction.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability of selecting one red marble and two blue marbles, we first determine the total number of marbles, which is 10 red + 6 blue = 16 marbles. 

Next, we calculate the total number of ways to choose 3 marbles from 16. This is given by the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, and k is the number of items to choose. Thus, the total combinations of 3 marbles from 16 is C(16, 3) = 16! / (3!(16-3)!) = 560.

Now, we need to find the number of favorable outcomes for selecting 1 red and 2 blue marbles. The number of ways to choose 1 red marble from 10 is C(10, 1) = 10. The number of ways to choose 2 blue marbles from 6 is C(6, 2) = 6! / (2!(6-2)!) = 15.

The total number of favorable outcomes for selecting 1 red and 2 blue marbles is the product of the two combinations: 10 (ways to choose red) * 15 (ways to choose blue) = 150.

Finally, the probability of selecting 1 red and 2 blue marbles is the ratio of the number of favorable outcomes to the total outcomes: P(1 red, 2 blue) = 150 / 560. 

To simplify this fraction, we find the greatest common divisor (GCD) of 150 and 560, which is 10. Dividing both the numerator and the denominator by 10 gives us 15 / 56.

Thus, the final answer is 15/56.<|END_OF_LATENT|> The final answer is $\frac{15}{56}$.