Consider events A and \( \mathrm{B} . P(A)=0,43, P(B)=0,37 \) and \( P(A \) and \( B)=0,1591 \). Determine whether events A and B are: \( \begin{array}{ll}1.1 \text { complementary } & \text { (2) } \\ 1.2 \text { exhausted } & \text { (2) } \\ 1.3 \text { mutually exclusive } & \text { (2) } \\ 1.4 \text { independent. } & \text { (2) } \\ 1.5 \text { Draw a Venn diagram to represent the given information. }\end{array} \)

To analyze the relationship between events A and B, we will use the provided probabilities: - \( P(A) = 0.43 \) - \( P(B) = 0.37 \) - \( P(A \cap B) = 0.1591 \) ### 1.1 Complementary Events Two events A and B are complementary if \( P(A) + P(B) = 1 \). Calculating: \[ P(A) + P(B) = 0.43 + 0.37 = 0.80 \] Since \( 0.80 \neq 1 \), events A and B are **not complementary**. ### 1.2 Exhausted Events Events A and B are said to be exhaustive if their union covers the entire sample space, i.e., \( P(A \cup B) = 1 \). Using the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Calculating: \[ P(A \cup B) = 0.43 + 0.37 - 0.1591 = 0.6309 \] Since \( 0.6309 \neq 1 \), events A and B are **not exhaustive**. ### 1.3 Mutually Exclusive Events Events A and B are mutually exclusive if \( P(A \cap B) = 0 \). Since \( P(A \cap B) = 0.1591 \), events A and B are **not mutually exclusive**. ### 1.4 Independent Events Events A and B are independent if: \[ P(A \cap B) = P(A) \cdot P(B) \] Calculating \( P(A) \cdot P(B) \): \[ P(A) \cdot P(B) = 0.43 \cdot 0.37 = 0.1591 \] Since \( P(A \cap B) = 0.1591 \) matches \( P(A) \cdot P(B) \), events A and B are **independent**. ### 1.5 Venn Diagram To represent the information in a Venn diagram: 1. Draw two overlapping circles, one for event A and one for event B. 2. Label the area where both circles overlap with \( P(A \cap B) = 0.1591 \). 3. The area of circle A (not overlapping with B) will be \( P(A) - P(A \cap B) = 0.43 - 0.1591 = 0.2709 \). 4. The area of circle B (not overlapping with A) will be \( P(B) - P(A \cap B) = 0.37 - 0.1591 = 0.2109 \). 5. The area outside both circles will be \( 1 - P(A \cup B) = 1 - 0.6309 = 0.3691 \). The Venn diagram will look like this: ``` +-------------------+ | | | A | | 0.2709 | | | | +---------+ | | | | | | | 0.1591| | | | | | | +---------+ | | B | | 0.2109 | | | +-------------------+ ``` In summary: - Events A and B are **not complementary**. - Events A and B are **not exhaustive**. - Events A and B are **not mutually exclusive**. - Events A and B are **independent**.