https://file.aiquickdraw.com/mathbot/file/1726348784224b6lsth8x.jpg

Let's go through each of the questions in the image.

### Question 28:
We are given:
- \( P(A) = 0.8 \)
- \( P(B) = 0.2 \)
- \( P(A \cap B) = 0.16 \)

We need to determine the relationship between events \( A \) and \( B \).

1. **Check if \( A \) and \( B \) are independent**:
   Two events \( A \) and \( B \) are independent if:
   \[
   P(A \cap B) = P(A) \times P(B)
   \]
   Let's check:
   \[
   P(A) \times P(B) = 0.8 \times 0.2 = 0.16
   \]
   Since \( P(A \cap B) = 0.16 \), \( A \) and \( B \) are independent.

2. **Check if \( A \) and \( B \) are disjoint**:
   Two events are disjoint if \( P(A \cap B) = 0 \). Since \( P(A \cap B) = 0.16 \neq 0 \), the events are **not disjoint**.

So, the correct answer is:
- **B) Independent but not disjoint**.

### Question 29:
We are asked to identify the sample points in the event \( A \cap B \).

- \( A \) is the event where the number is even: \( A = \{2, 4, 6, 8, 10\} \).
- \( B \) is the event where the number is less than 7: \( B = \{1, 2, 3, 4, 5, 6\} \).

The event \( A \cap B \) represents the numbers that are in both \( A \) and \( B \). So, we need to find the intersection of the two sets:
\[
A \cap B = \{2, 4, 6\}
\]

Therefore, the correct answer is:
- **C) \{2, 4, 6\}**.

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Would you like further details or explanations on any part of these solutions? Here are 5 related questions that you could explore:

1. How do you test if two events are disjoint using probabilities?
2. What is the difference between disjoint and independent events?
3. How do you calculate the intersection of two events with known probabilities?
4. If \( P(A) = 0.8 \), what is the probability of the complement of event \( A \)?
5. How does the concept of mutually exclusive events apply to real-life situations?

**Tip**: Always double-check whether two events are independent by comparing \( P(A \cap B) \) to \( P(A) \times P(B) \).

Question 28: Select the most appropriate answer.
For two events A and B, P(A) = 0.8, P(B) = 0.2, and P(A and B) = 0.16. It follows that A and B are:
A) disjoint but not independent.
B) independent but not disjoint.
C) complementary.
D) neither disjoint nor independent.
E) both disjoint and independent.

Question 29: Solve the problem.
A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows:
A: {The number is even}.
B: {The number is less than 7}.
Identify the sample points in the event A ∩ B.
A) {1, 2, 3, 4, 5, 6, 7, 8, 10}
B) {1, 2, 3, 4, 5, 6, 8, 10}
C) {2, 4, 6}
D) {1, 2, 3, 4, 5, 6, 7, 9}

This article solves two probability questions. The first asks to determine if two events A and B are independent or disjoint with given probabilities. The second identifies sample points for the intersection of two events, A (numbers even) and B (numbers less than 7). Suitable for students in Grade 11-12.

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