"[NW] Going to Disney World John, Roberto, Clarice, Dominique, and Marco work for a publishing company. The company wants to send two employees to a statistics conference in Orlando. To be fair, the company decides that the two individuals who get to attend will have their names randomly drawn from a hat.


d. What is the probability that John stays home?"

Verified step by step guidance

1. Identify the total number of employees eligible for selection. In this case, there are 5 employees: John, Roberto, Clarice, Dominique, and Marco.
2. Determine the total number of ways to choose 2 employees out of 5. This is a combination problem, so use the formula for combinations: \(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n=5\) and \(k=2\).
3. Calculate the total number of possible pairs that can be drawn from the 5 employees using the combination formula: \(C(5, 2)\).
4. Next, find the number of favorable outcomes where John is NOT selected. This means both selected employees come from the other 4 employees (Roberto, Clarice, Dominique, and Marco). Calculate the number of ways to choose 2 employees from these 4 using \(C(4, 2)\).
5. Finally, compute the probability that John stays home by dividing the number of favorable outcomes (John not selected) by the total number of possible outcomes (any 2 employees selected): \(P(\text{John stays home}) = \frac{C(4, 2)}{C(5, 2)}\).

Key Concepts

Sample Space

The sample space is the set of all possible outcomes in a probability experiment. In this case, it includes all possible pairs of employees that can be chosen to attend the conference. Understanding the sample space helps determine the total number of equally likely outcomes.

Combinations

Combinations refer to the selection of items where order does not matter. Since the company is choosing 2 employees out of 5 without regard to order, the number of possible pairs is calculated using combinations, specifically "5 choose 2".

Complement Rule

The complement rule states that the probability of an event not happening is one minus the probability that it does happen. To find the probability that John stays home, we can find the probability that John is chosen and subtract it from 1.