Question

2.2 Given: P(A) = 0, 3
P(B) = 0, 4
P(A or B) = 0, 7 where A and B are two different events
2.2.1 Calculate P(A and B). (3)
2.2.2 Are events A and B independent? Show your calculations. (4)
2.3 Classify the following events as mutually exclusive, independent or
neither.
2.3.1 Rolling an odd number on a die and rolling an even number
on a die. (2)
2.3.2 Rain falling in South Africa and winning the lottery in Spain. (2)
[19]

Answer 1

To solve the problems step by step, we will first address question 2.2 regarding the probabilities of events A and B.

### 2.2.1 Calculate P(A and B)

We can use the formula for the probability of the union of two events:

\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\]

Given:
- \( P(A) = 0.3 \)
- \( P(B) = 0.4 \)
- \( P(A \text{ or } B) = 0.7 \)

Plugging in the values we have:

\[
0.7 = 0.3 + 0.4 - P(A \text{ and } B)
\]

Now, solve for \( P(A \text{ and } B) \):

\[
0.7 = 0.7 - P(A \text{ and } B)
\]

Rearranging gives:

\[
P(A \text{ and } B) = 0.7 - 0.7 = 0
\]

So, we find:

\[
P(A \text{ and } B) = 0
\]

### 2.2.2 Are events A and B independent? Show your calculations.

Two events A and B are considered independent if:

\[
P(A \text{ and } B) = P(A) \times P(B)
\]

From our previous calculation, we found \( P(A \text{ and } B) = 0 \).

Now, we calculate \( P(A) \times P(B) \):

\[
P(A) \times P(B) = 0.3 \times 0.4 = 0.12
\]

Since \( P(A \text{ and } B) \neq P(A) \times P(B) \) (0 is not equal to 0.12), events A and B are not independent.

### Summary of 2.2
- \( P(A \text{ and } B) = 0 \)
- Events A and B are **not independent** since \( P(A \text{ and } B) \neq P(A) \times P(B) \).

---

### 2.3 Classify the following events:

#### 2.3.1 Rolling an odd number on a die and rolling an even number on a die.

- **Mutually Exclusive:** Two events are mutually exclusive if they cannot happen at the same time. Since you cannot roll an odd number and an even number on a single roll of a die at the same time, these events are mutually exclusive.

**Classification:** **Mutually Exclusive**

#### 2.3.2 Rain falling in South Africa and winning the lottery in Spain.

- **Independent:** Events are independent if the occurrence of one does not affect the occurrence of the other. Rain in South Africa and winning the lottery in Spain are unrelated events; the outcome of one does not impact the other.

**Classification:** **Independent**

### Summary of 2.3
- 2.3.1: **Mutually Exclusive**
- 2.3.2: **Independent**