There are 7 girls and 6 boys in a class. Suppose five children are randomly chosen to decorate the game room. Given that at least one boy is chosen, what is the probability that exactly two boys are chosen?

It should be 5 over 6. Which equal to 1 over 2.

Hope it help.

To find the probability that exactly two boys are chosen given that at least one boy is chosen, we need to first calculate the total number of possible outcomes and the favorable outcomes.

Total number of outcomes:
Since there are 7 girls and 6 boys in the class, the total number of children is 13. We need to choose 5 children from these 13, which can be done in C(13, 5) ways using combinations.

Favorable outcomes:
We can calculate the number of favorable outcomes in two steps:
Step 1: Calculate the number of ways to choose exactly two boys from the 6 boys available. This can be done in C(6, 2) ways.
Step 2: Choose the remaining 3 children from the remaining pool of boys and girls. We need to subtract the cases where all 3 remaining children are girls since we want at least one boy to be chosen.

Now let's calculate the probability.

Total number of outcomes = C(13, 5) = 1287
Number of favorable outcomes = C(6, 2) * (C(7, 3) - C(7, 0)) = 15 * (35 - 1) = 510

Probability = Number of favorable outcomes / Total number of outcomes
Probability = 510 / 1287 ≈ 0.396

Therefore, the probability that exactly two boys are chosen given that at least one boy is chosen is approximately 0.396.