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Chapter 6: Problem 28

A student placement center has requests from five students for interviews regarding employment with a particular consulting firm. Three of these students are math majors, and the other two students are statistics majors. Unfortunately, the interviewer has time to talk to only two of the students. These two will be randomly selected from among the five. a. What is the probability that both selected students are statistics majors? b. What is the probability that both students are math majors? c. What is the probability that at least one of the students selected is a statistics major? d. What is the probability that the selected students have different majors?

Short Answer

Expert verified
a. The probability that both students selected are statistics majors is 0.1. b. The probability that both students are math majors is 0.3. c. The probability that at least one student selected is a Statistics major is 0.7. d. The probability that the selected students have different majors is 0.6.

Step by step solution

01

Total number of combinations

The total number of ways of choosing 2 students out of 5 regardless of their majors is given by the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), where n is the total number of students (5) and k is the number of students to be chosen (2). So, \(C(5, 2) = \frac{5!}{2!(5-2)!}=10\).
02

Probability that both selected students are statistics majors

The number of ways we can choose two statistics students out of two is \(C(2, 2) = \frac{2!}{2!(2-2)!}=1\). So, the probability is given by the ratio of this to the total number of combinations, which is \( \frac{1}{10} = 0.1\).
03

Probability that both students are math majors

The number of ways we can choose two math students out of three is \(C(3, 2) = \frac{3!}{2!(3-2)!}=3\). The probability is thus \( \frac{3}{10} = 0.3\).
04

Probability that at least one of the students selected is a Statistics major

The only case where there isn't at least one statistics major is when both students are math majors, and this probability we already calculated as 0.3. Thus, the probability that at least one is a statistics major is \(1 - Pr(\text{both are math majors}) = 1 - 0.3 = 0.7\).
05

Probability that the selected students have different majors

This occurs if one student is a math major and the other is a statistics major. The number of ways to choose one math major out of three is \(C(3, 1) = 3\), and the number of ways to choose one statistics major out of two is \(C(2, 1) = 2\). So the total number of ways of choosing one from each major is \(3*2 = 6\). The probability is thus \( \frac{6}{10} = 0.6\).

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