Probability of Selecting At Least One Red Marble

A bag contains 5 red, 6 blue, and 4 green marbles. If you randomly select 3 marbles from the bag without replacement, what is the probability that at least one marble is red?

To solve this problem, we can use the concept of complementary probability. Instead of finding the probability of getting at least one red marble directly, we will calculate the probability of not getting any red marbles and subtract that from 1.

The total number of marbles is: $$5 + 6 + 4 = 15$$.

The probability of selecting only blue and green marbles (not red) can be calculated as follows:

First, we find the number of ways to choose 3 marbles from the 10 non-red marbles (6 blue + 4 green).

Then, we calculate the total ways to choose 3 marbles from all 15. The probability is the ratio of these two: $$P(\text{not red}) = \f\frac{\text{Ways to choose 3 from 10}}{\text{Ways to choose 3 from 15}}$$.

Finally, compute the required probability as:

$$P(\text{at least one red}) = 1 - P(\text{not red})$$.