Question

Weekdays 34 25 59
Weekends 56 65 121
Total 90 90 180
Use the table to answer the following questions. Round your answers to the nearest percent.

Let event A be a person likes tigers, and event B be a person prefers weekdays.
First, find the probability of a person liking tigers, given that they prefer weekdays. P(A|B) is about
%. Second, find the probability of a person preferring weekdays, given they like tigers. P(B|A) is about
%.

Answer 1

To find the probabilities \( P(A|B) \) and \( P(B|A) \), we will use the definitions of conditional probabilities.

### Step 1: Identify the total numbers

From the table:

- Total number of respondents: \( 180 \)
- Total who prefer weekdays: \( 90 \)
- Total who like tigers: To find this, we need to sum the numbers given in the total row (that are associated with tigers).

From the given data (assuming the first column is the number that likes tigers):
- Number who like tigers and prefer weekdays: \( 34 \)
- Number who like tigers and prefer weekends: \( 56 \)
- Total who like tigers: \( 34 + 56 = 90 \)

### Step 2: Calculate \( P(A|B) \)

\( P(A|B) \) is the probability that a person likes tigers given that they prefer weekdays.

Using the formula for conditional probability:
\[
P(A|B) = \frac{P(A \cap B)}{P(B)}
\]
Where:
- \( P(A \cap B) \) (those who like tigers and prefer weekdays) is \( 34 \).
- \( P(B) \) (total who prefer weekdays) is \( 90 \).

Thus:
\[
P(A|B) = \frac{34}{90} \approx 0.3778
\]

Converted to a percentage, this is:
\[
P(A|B) \approx 0.378 \times 100 \approx 38\%
\]

### Step 3: Calculate \( P(B|A) \)

\( P(B|A) \) is the probability that a person prefers weekdays given that they like tigers.

Using the same formula:
\[
P(B|A) = \frac{P(A \cap B)}{P(A)}
\]
Where:
- \( P(A \cap B) \) (those who like tigers and prefer weekdays) is \( 34 \).
- \( P(A) \) (total who like tigers) is \( 90 \).

Thus:
\[
P(B|A) = \frac{34}{90} \approx 0.3778
\]

Converted to a percentage, this is:
\[
P(B|A) \approx 0.378 \times 100 \approx 38\%
\]

### Summary of Results:
- \( P(A|B) \) is about **38%**.
- \( P(B|A) \) is about **38%**.