In a bag are three red marbles and three green marbles. If two marbles are drawn from the bag at the same time, what is the probability that both marbles will be red?

prob(both red) = (3/6)(2/5) = 1/5

To find the probability of drawing two red marbles, we need to determine the total number of outcomes and the number of favorable outcomes.

First, let's find the total number of outcomes. When two marbles are drawn at the same time, there are C(6, 2) ways to choose two marbles from a total of six marbles. The notation C(n,r) represents the number of combinations of r items chosen from a set of n items, and can be calculated using the formula C(n,r) = n! / (r! * (n-r)!)

So, C(6,2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5 * 4!) / (2! * 4!) = (6 * 5) / 2 = 15

Therefore, there are 15 possible outcomes when two marbles are drawn from the bag simultaneously.

Next, let's determine the number of favorable outcomes. We want to draw two red marbles, and there are three red marbles in the bag. So, we need to choose two red marbles from the three available. This can be calculated using C(3,2) = 3! / (2! * (3-2)!) = 3! / (2! * 1!) = (3 * 2!) / 2 = 3

Therefore, there are 3 favorable outcomes where both marbles drawn will be red.

Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:

Probability = favorable outcomes / total outcomes = 3 / 15 = 1/5 = 0.2

Therefore, the probability that both marbles drawn will be red is 0.2, or 20%.