Question

math 2500, final exam
december 10, 2025; 11:00 am - 1:00 pm (2 hours)
name:
instructions. show all your work to get full credit. circle your final answers.
the exam is worth 40 points that is 20% of your final grade.

1. (3 points) the probability that a tourist traveling to europe will visit paris is.8, the probability that the tourist will visit rome is.6, and the probability that the tourist will visit either of the cities is.9. find the probability that the tourist will visit:

a) both cities.
b) neither city.

Explanation:

Part (a)

Step 1: Recall the formula for the probability of the union of two events

The formula for the probability of the union of two events \( A \) and \( B \) is \( P(A \cup B)=P(A)+P(B)-P(A \cap B) \), where \( P(A \cup B) \) is the probability of either event \( A \) or event \( B \) occurring, \( P(A) \) is the probability of event \( A \) occurring, \( P(B) \) is the probability of event \( B \) occurring, and \( P(A \cap B) \) is the probability of both events \( A \) and \( B \) occurring.

Let \( A \) be the event that a tourist visits Paris and \( B \) be the event that a tourist visits Rome. We know that \( P(A) = 0.8 \), \( P(B)=0.6 \), and \( P(A \cup B)=0.9 \). We need to find \( P(A \cap B) \).

Step 2: Rearrange the formula to solve for \( P(A \cap B) \)

From \( P(A \cup B)=P(A)+P(B)-P(A \cap B) \), we can rearrange it to \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \).

Substitute the given values: \( P(A \cap B)=0.8 + 0.6-0.9 \).

First, calculate \( 0.8+0.6 = 1.4 \). Then, subtract \( 0.9 \) from \( 1.4 \): \( 1.4 - 0.9=0.5 \).

Part (b)

Step 1: Recall the formula for the probability of the complement of the union of two events

The probability of neither event \( A \) nor event \( B \) occurring is the complement of the event that either \( A \) or \( B \) occurs. That is, \( P(\text{neither } A \text{ nor } B)=1 - P(A \cup B) \).

Step 2: Substitute the value of \( P(A \cup B) \)

We know that \( P(A \cup B) = 0.9 \). So, \( P(\text{neither } A \text{ nor } B)=1 - 0.9 = 0.1 \).

Answer:

a) The probability that the tourist will visit both cities is \( \boldsymbol{0.5} \).

b) The probability that the tourist will visit neither city is \( \boldsymbol{0.1} \).