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### **Question 4.1**

Given:
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{5} \)

If A and B are mutually exclusive events, we need to determine:

#### **4.1.1: \( P(A \text{ and } B) \)**
For mutually exclusive events, the probability of both events occurring together is zero:

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

#### **4.1.2: \( P(A \text{ or } B) \)**
The formula for the union of two mutually exclusive events is:

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

Substitute the given values:

\[
P(A \text{ or } B) = \frac{1}{2} + \frac{1}{5} = \frac{5}{10} + \frac{2}{10} = \frac{7}{10} = 0.7
\]

### **Question 4.2**

Given:
- Events A and B are mutually exclusive.
- \( P(B') = 0.4 \)
- \( P(A \cup B) = 0.7 \)

We need to calculate \( P(A) \).

Since events A and B are mutually exclusive, we have:

\[
P(A \cup B) = P(A) + P(B)
\]

Let \( P(B) = 1 - P(B') \):

\[
P(B) = 1 - 0.4 = 0.6
\]

Now, substitute into the union formula:

\[
0.7 = P(A) + 0.6 \implies P(A) = 0.7 - 0.6 = 0.1
\]

### **Question 4.3**

Given:
- \( P(A) = 0.30 \)
- \( P(B) = 0.65 \)
- \( P(A \cup B) = 0.74 \)

We need to calculate \( P(A \text{ and } B) \).

For any two events, the probability of their union is given by:

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

Rearrange to solve for \( P(A \text{ and } B) \):

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

Substitute the given values:

\[
P(A \text{ and } B) = 0.30 + 0.65 - 0.74 = 0.95 - 0.74 = 0.21
\]

So, \( P(A \text{ and } B) = 0.21 \).

### **Drawing the Venn Diagram:**

To represent this information visually:

1. Draw two overlapping circles representing events A and B.
2. The overlap (intersection) should be labeled as \( P(A \text{ and } B) = 0.21 \).
3. The portion of circle A outside of the intersection is \( P(A) - P(A \text{ and } B) = 0.30 - 0.21 = 0.09 \).
4. The portion of circle B outside of the intersection is \( P(B) - P(A \text{ and } B) = 0.65 - 0.21 = 0.44 \).

Would you like further details on any part, or help with another question?

### **Related Questions:**

1. What is the difference between mutually exclusive and independent events?
2. How do you calculate the probability of the union of three events?
3. What are complementary events, and how are they calculated?
4. How can you represent more than two events in a Venn diagram?
5. How do you calculate conditional probability when given \( P(A | B) \)?

### **Tip:**
Always double-check your calculations by verifying that the sum of all probabilities in a given problem equals 1.

A and B are two events with P(A) = 1/2 and P(B) = 1/5. If A and B are mutually exclusive, determine P(A and B) and P(A or B). In other parts, you are asked to calculate P(A) for given conditions and to find P(A and B) based on additional probabilities.

Learn how to solve probability problems involving mutually exclusive events. This problem walks you through calculating P(A and B) and P(A or B) with given values like P(A) = 1/2 and P(B) = 1/5. The guide also includes a detailed explanation of the union and intersection of events.

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