Question

Six people hope to be selected as a contestant on a TV game show. Two of these people are younger than 25 years old. Two of these six will be chosen at random to be on the show. a. What is the sample space for the chance experiment of selecting two of these people at random? (Hint: You can think of the people as being labeled \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E},\) and \(\mathrm{F}\). One possible selection of two people is \(\mathrm{A}\) and \(\mathrm{B}\). There are 14 other possible selections to consider.) b. Are the outcomes in the sample space equally likely? c. What is the probability that both the chosen contestants are younger than \(25 ?\) d. What is the probability that both the chosen contestants are not younger than \(25 ?\) e. What is the probability that one is younger than 25 and the other is not?

Short answer

a. There are 15 possible contestant pairs. b. Yes, all outcomes are equally likely. c. The probability that both chosen contestants are younger than 25 is \(\frac{1}{15}\). d. The probability that both chosen contestants are not younger than 25 is \(\frac{2}{5}\). e. The probability that one contestant is younger than 25 and the other is not is \(\frac{8}{15}\).

Step-by-step solution

Step 1. Determine the Sample Space

The sample space is the set of all possible pairs of contestants that could be chosen from the group of six. Since the order does not matter (i.e., AB is the same as BA), we look for combinations of 2 out of 6. Use the combination formula \(\text{C}(n, k) = \frac{n!}{k!(n-k)!}\), where \(n!\) denotes the factorial of \(n\), \(k!\) denotes the factorial of \(k\), and \((n-k)!\) denotes the factorial of \((n-k)\). Based on this formula, there are \(\text{C}(6, 2) = 15\) possible contestant pairs, so the sample space consists of 15 pairs.

Step 2. Evaluate the Likelihood of Each Outcome

In this randomly selected process, each pair of contestants has an equivalent opportunity to be chosen. So, yes, the outcomes in the sample space are equally likely.

Step 3. Calculate the Probability of Selecting Two Contestants Younger than 25

There are two contestants who are younger than 25. So, when selecting two contestants, the number of successful outcomes where both are young will be given by \(\text{C}(2, 2) = 1\). So the probability will be the ratio of successful outcomes to total outcomes, which is \(\frac{1}{15}\).

Step 4. Compute the Probability of Selecting Two Contestants Not Younger than 25

There are four contestants who are not younger than 25. The number of successful outcomes for picking two older contestants will be given by \(\text{C}(4, 2) = 6\). So the probability will be the ratio of successful outcomes to total outcomes, which is \(\frac{6}{15}\), simplified to \(\frac{2}{5}\).

Step 5. Find the Probability of Selecting One Young and One Old Contestant

If the two selected contestants include one who is younger than 25 and one who is not, we would select one out of the two young contestants and one out of the four older contestants. So, the number of successful outcomes will be given by \(\text{C}(2, 1) \times \text{C}(4, 1) = 2 \times 4 = 8\). So the probability will be the ratio of successful outcomes to total outcomes, which is \(\frac{8}{15}\).