Question

Students are surveyed about art styles they prefer. The results are shown below.

Likes Painting Does Not Like Painting Total
Likes Pottery 26 7 33
Does Not Like Pottery 31 34 65
Total 57 41 98
Let A be the event "a student does not like painting" and B be the event "a student likes pottery." Determine if A and B are independent events.

Choose the correct answers from the lists.

(2 points)
P(A|B)≈

and P(A)≈


So, the events are
.

Answer 1

To determine if events A ("a student does not like painting") and B ("a student likes pottery") are independent, we need to calculate the probabilities \( P(A|B) \) and \( P(A) \) and compare them.

### Step 1: Calculate \( P(A) \)

\[
P(A) = \frac{\text{Number of students who do not like painting}}{\text{Total number of students}}
\]

From the table:
- Number who do not like painting = 41
- Total number of students = 98

\[
P(A) = \frac{41}{98} \approx 0.4184
\]

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

\[
P(B) = \frac{\text{Number of students who like pottery}}{\text{Total number of students}}
\]

From the table:
- Number who like pottery = 33
- Total number of students = 98

\[
P(B) = \frac{33}{98} \approx 0.3367
\]

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

This is the probability that a student does not like painting and likes pottery. From the table:

- Number of students who like pottery and do not like painting = 7

\[
P(A \cap B) = \frac{7}{98} \approx 0.0714
\]

### Step 4: Calculate \( P(A|B) \)

\[
P(A|B) = \frac{P(A \cap B)}{P(B)}
\]

\[
P(A|B) = \frac{\frac{7}{98}}{\frac{33}{98}} = \frac{7}{33} \approx 0.2121
\]

### Step 5: Check for independence

Events A and B are independent if \( P(A|B) = P(A) \).

We have:

- \( P(A|B) \approx 0.2121 \)
- \( P(A) \approx 0.4184 \)

Since \( P(A|B) \neq P(A) \), the events A and B are **not independent**.

### Final Answers

- \( P(A|B) \approx 0.2121 \)
- \( P(A) \approx 0.4184 \)

So, the events are **not independent**.