Suppose that a certain college class contains 39 students. Of these, 24 are seniors, 21 are mathematics majors, and 5 are neither. A student is selected at random from the class.(a) What is the probability that the student is both a senior and a mathematics major?(b) Given that the student selected is a senior, what is the probability that she is also a mathematics major?

To solve this problem, we first need to understand the total number of students, the number of seniors, the number of mathematics majors, and the number of students who are neither.

From the problem, we know that:

• The total number of students in the class is 39.
• The number of seniors is 24.
• The number of mathematics majors is 21.
• The number of students who are neither seniors nor mathematics majors is 5.

(a) To find the probability that a student selected at random is both a senior and a mathematics major, we first need to find the number of students who are both seniors and mathematics majors.

We can find this by adding the number of seniors and the number of mathematics majors and then subtracting the total number of students and the number of students who are neither. This is because the total number of students includes those who are both seniors and mathematics majors, so we need to subtract the total number of students to avoid double-counting. We also need to add back in the number of students who are neither, because we subtracted them out when we subtracted the total number of students.

So, the number of students who are both seniors and mathematics majors is 24 (seniors) + 21 (mathematics majors) - 39 (total students) + 5 (neither) = 11.

The probability that a student selected at random is both a senior and a mathematics major is then 11 (both) / 39 (total) = 0.282.

(b) To find the probability that a student selected at random is a mathematics major given that she is a senior, we need to find the number of seniors who are mathematics majors and divide by the total number of seniors.

From part (a), we know that the number of students who are both seniors and mathematics majors is 11. So, the probability that a student selected at random is a mathematics major given that she is a senior is 11 (both) / 24 (seniors) = 0.458.

To solve this problem, we first need to understand the total number of students, the number of seniors, the number of mathematics majors, and the number of students who are neither.

From the problem, we know that:

• The total number of students in the class is 39.
• The number of seniors is 24.
• The number of mathematics majors is 21.
• The number of students who are neither seniors nor mathematics majors is 5.

(a) To find the probability that a student selected at random is both a senior and a mathematics major, we first need to find the number of students who are both seniors and mathematics majors.

We can find this by adding the number of seniors and the number of mathematics majors and then subtracting the total number of students and the number of students who are neither. This is because the total number of students includes those who are both seniors and mathematics majors, so we need to subtract the total number of students to avoid double-counting. We also need to add back in the number of students who are neither, because we subtracted them out when we subtracted the total number of students.

So, the number of students who are both seniors and mathematics majors is 24 (seniors) + 21 (mathematics majors) - 39 (total students) + 5 (neither) = 11.

The probability that a student selected at random is both a senior and a mathematics major is then 11 (both) / 39 (total) = 0.282.

(b) To find the probability that a student selected at random is a mathematics major given that she is a senior, we need to find the number of seniors who are mathematics majors and divide by the total number of seniors.

From part (a), we know that the number of students who are both seniors and mathematics majors is 11. So, the probability that a student selected at random is a mathematics major given that she is a senior is 11 (both) / 24 (seniors) = 0.458.