Question

In a high school statistics class, the teacher formed 10 groups of project teams, each consisting of 3 girls and 2 boys. The teacher is randomly selecting two presenters of the results from each group. If X is the random variable of the number of teams having only girl presenters, what is E(X)? Assume that no student is in more than one team.

Answer 1

The expected value of the number of teams with only girl presenters, when choosing 2 presenters from 10 groups each consisting of 3 girls and 2 boys, is 2.25.

Step-by-step explanation:

The question asks us to find the expected value (E(X)) of the number of teams with only girl presenters when groups of 5 students (3 girls and 2 boys) are formed, and 2 presenters are chosen randomly from each group. We must calculate the expected value based on the probability that both selected presenters are girls. Since there are 10 groups, the expected value will be the sum of these probabilities over all groups.

To calculate the probability of selecting 2 girls from a single group, we use combinations:

P(2 girls) = C(3, 2) / C(5, 2)

Plugging in the values, we get:

P(2 girls) = (3! / (2! * (3 - 2)!)) / (5! / (2! * (5 - 2)!))

Therefore, for one team:

P(2 girls) = (3 / (2 * 1)) / (10 / (2 * 3))

P(2 girls) = 9/20

Since the random variable X counts the number of teams with only girls as presenters, we must consider this probability for each of the 10 groups.

The expected value for X across all groups is:

E(X) = 10 * (9/20)

E(X) = 45/20

E(X) = 2.25

Thus, the expected value of the number of teams having only girl presenters is 2.25.