Question

A high school science teacher has 78 students. Of those students, 35 are in the band and 32 are on a sports team. There are 16 students who are not in the band or on a sports team. One student from the 78 students will be selected at random. Let event B represent the event of selecting a student in the band, and let event S represent the event of selecting a student on a sports team.

Are B and S mutually exclusive events?

A) No, because P(B∩S) = 5/78.
B) No, because P(B∩S) = 4/78.
C) No, because P(B∩S) = 6/78.
D) Yes, because P(B∩S) = 5/78.
E) Yes, because P(B∩S) = 6/78.

Answer 1

Option (A), The two events B and S are not mutually exclusive since there is an overlap of students who are in both the band and on a sports team, giving a probability of 5/78 for a student to be in both.

Step-by-step explanation:

The events B and S are not mutually exclusive if there are students who are both in the band and on a sports team. To find out if B and S are mutually exclusive, we need to calculate P(B∩S), the probability that a student is both in the band and on a sports team.

The total number of students is 78.

There are 35 in the band, 32 on a sports team, and 16 that are neither.

So, we subtract the 16 from the total to find the number that are in either of the groups or both: 78 - 16 = 62.

Now we subtract this number from the sum of students in the band and on sports teams to find those who are in both groups: 35 + 32 - 62 = 5.

Thus, the probability that a randomly selected student is in both groups is 5/78.

Therefore, the correct answer is A) No, because P(B∩S) = 5/78.