Question

A student placement center has requests from five students for employment interviews. 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 sample space for the chance experiment of selecting two students at random? (Hint: You can think of the students as being labeled \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},\) and \(\mathrm{E}\). One possible selection of two students is \(\mathrm{A}\) and \(\mathrm{B}\). There are nine other possible selections to consider.) b. Are the outcomes in the sample space equally likely? c. What is the probability that both selected students are statistics majors? d. What is the probability that both students are math majors? e. What is the probability that at least one of the students selected is a statistics major? f. What is the probability that the selected students have different majors?

Short answer

\[ P(\text{two statistics majors}) = \frac{1}{10}, P(\text{two math majors}) = \frac{3}{10}, P(\text{at least one statistics major}) = \frac{1}{2}, P(\text{students with different majors}) = \frac{3}{5} \]

Step-by-step solution

Define the sample space

Label the students as A, B, C, D and E, where A, B, C are math majors and D, E are statistics majors. The sample space of selecting two students at random can be defined as {AB, AC, AD, AE, BC, BD, BE, CD, CE, DE}, where AB represents selecting student A and student B.

Evaluate equal likelihood of outcomes

Since each pair of students is selected randomly, all outcomes in the sample space are indeed equally likely.

Calculate the probability of selecting two statistics majors

The favorable outcomes for this event are {DE}, which is 1 out of 10 total outcomes. Hence, the probability is \(\frac{1}{10}\).

Calculate the probability of selecting two math majors

The favorable outcomes for this event are {AB, AC, BC}, which are 3 out of 10 total outcomes. Hence, the probability is \(\frac{3}{10}\).

Calculate the probability of selecting at least one statistics major

The favorable outcomes for this event are {AD, AE, BD, BE, DE}, which are 5 out of 10 total outcomes. Hence, the probability is \(\frac{5}{10} = \frac{1}{2}\).

Calculate the probability of selecting students with different majors

The favorable outcomes for this event are {AD, AE, BD, BE, CD, CE}, which are 6 out of 10 total outcomes. Hence, the probability is \(\frac{6}{10} = \frac{3}{5}\).