Question

1. clarice, john, marco, and robert work for a publishing company. the company wants to send two employees to a statistics conference. to be fair, the company decides that the two individuals who get to attend will have their names randomly drawn from a hat. a. determine the sample space of the experiment. b. what is the probability that clarice and john attend the conference? c. what is the probability that john attends the conference? d. what is the probability that clarice stays home?

Explanation:

Step1: Find sample - space using combination

The number of ways to choose 2 out of 4 employees is given by the combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 4$ and $k=2$. Also, we can list out the pairs. The sample space $S=\{(Clarice, John),(Clarice, Marco),(Clarice, Robert),(John, Marco),(John, Robert),(Marco, Robert)\}$.

Step2: Calculate probability for part b

The probability $P$ of an event $E$ is $P(E)=\frac{n(E)}{n(S)}$. For the event $E$ that Clarice and John attend, $n(E) = 1$ and $n(S)=6$. So $P=\frac{1}{6}$.

Step3: Calculate probability for part c

The event that John attends consists of the pairs $\{(Clarice, John),(John, Marco),(John, Robert)\}$. So $n(E)=3$ and $n(S) = 6$. Then $P=\frac{3}{6}=\frac{1}{2}$.

Step4: Calculate probability for part d

The event that Clarice stays home means the pairs do not contain Clarice. The pairs are $\{(John, Marco),(John, Robert),(Marco, Robert)\}$. So $n(E)=3$ and $n(S)=6$. Then $P=\frac{3}{6}=\frac{1}{2}$.

Answer:

a. $S=\{(Clarice, John),(Clarice, Marco),(Clarice, Robert),(John, Marco),(John, Robert),(Marco, Robert)\}$
b. $\frac{1}{6}$
c. $\frac{1}{2}$
d. $\frac{1}{2}$