There are two classrooms. The first classroom has 15 students, 5 of whom are girls and 10 are boys. The second classroom has 14 students, 6 of whom are girls and 8 are boys. 2 students move from the first classroom to the second classroom. Then, one student is called from the second classroom to the dean's office. What is the probability that the student called to the dean's office is a girl?

1. Problem Description

There are two classrooms. The first classroom has 15 students, 5 of whom are girls and 10 are boys. The second classroom has 14 students, 6 of whom are girls and 8 are boys. 2 students move from the first classroom to the second classroom. Then, one student is called from the second classroom to the dean's office. What is the probability that the student called to the dean's office is a girl?

2. Solution Steps

First, determine the composition of the first classroom after the transfer of students to the second classroom. We don't know the genders of the 2 students who moved. So, we must consider possible scenarios for the students moving from the first to the second classroom.

Scenario 1: Both students were girls.

Probability of this happening is $P(G_1G_2)$.

If two girls moved from the first classroom to the second, the first classroom has 3 girls and 10 boys. The second classroom has 6 + 2 = 8 girls and 8 boys, totaling 16 students.

The probability of choosing a girl from the second class is $8/16 = 1/2$.

Scenario 2: One girl and one boy moved.

If one girl and one boy moved from the first classroom to the second, the first classroom has 4 girls and 9 boys. The second classroom has 6 + 1 = 7 girls and 8 + 1 = 9 boys, totaling 16 students.

The probability of choosing a girl from the second class is $7/16$.

Scenario 3: Both students were boys.

If two boys moved from the first classroom to the second, the first classroom has 5 girls and 8 boys. The second classroom has 6 girls and 8 + 2 = 10 boys, totaling 16 students.

The probability of choosing a girl from the second class is $6/16 = 3/8$.

Now we calculate the probabilities of each scenario.

Let $G_1$ and $G_2$ be the events that the first and second students selected from the first classroom are girls, and $B_1$ and $B_2$ be the events that they are boys.

$P(G_1) = 5/15 = 1/3$

$P(B_1) = 10/15 = 2/3$

$P(G_2|G_1) = 4/14 = 2/7$

$P(B_2|G_1) = 10/14 = 5/7$

$P(G_2|B_1) = 5/14$

$P(B_2|B_1) = 9/14$

$P(G_1G_2) = P(G_1) * P(G_2|G_1) = (1/3) * (2/7) = 2/21$

$P(G_1B_2) = P(G_1) * P(B_2|G_1) = (1/3) * (5/7) = 5/21$

$P(B_1G_2) = P(B_1) * P(G_2|B_1) = (2/3) * (5/14) = 10/42 = 5/21$

$P(B_1B_2) = P(B_1) * P(B_2|B_1) = (2/3) * (9/14) = 18/42 = 3/7 = 9/21$

Probability (One girl and one boy) = $P(G_1B_2) + P(B_1G_2) = 5/21 + 5/21 = 10/21$

The probability of choosing a girl from the second classroom is:

$P(\text{Girl}) = P(G_1G_2) * (8/16) + P(\text{One girl, One boy}) * (7/16) + P(B_1B_2) * (6/16)$

$P(\text{Girl}) = (2/21) * (1/2) + (10/21) * (7/16) + (9/21) * (3/8)$

$P(\text{Girl}) = 1/21 + 70/336 + 27/168 = 1/21 + 5/24 + 9/56 = (8 + 35 + 27)/168 = 70/168 = 5/12$

3. Final Answer

5/12