Let's work through the problem step by step based on the information given in the image.

We are given:

• $P(A) = 0.4$
• $P(B) = 0.73$
• Events $A$ and $B$ are independent.

(a) $P(A \cap B)$ (The probability that both $A$ and $B$ occur)

Since $A$ and $B$ are independent, the probability of their intersection (both occurring) is given by: $P(A \cap B) = P(A) \times P(B) = 0.4 \times 0.73 = 0.2920$

(b) $P(A \cap B^c)$ (The probability that $A$ occurs and $B$ does not occur)

The complement rule states that $P(B^c) = 1 - P(B)$. Therefore: $P(B^c) = 1 - 0.73 = 0.27$ Now, since $A$ and $B^c$ are also independent: $P(A \cap B^c) = P(A) \times P(B^c) = 0.4 \times 0.27 = 0.1080$

(c) Probability that neither $A$ nor $B$ will occur

The probability that neither event occurs is given by $P(A^c \cap B^c)$. Since $A$ and $B$ are independent, we can compute: $P(A^c) = 1 - P(A) = 1 - 0.4 = 0.6$ $P(A^c \cap B^c) = P(A^c) \times P(B^c) = 0.6 \times 0.27 = 0.1620$

(d) Probability that exactly one of these two events will occur

This is the probability that either $A$ occurs and $B$ does not, or $B$ occurs and $A$ does not. This can be written as: $P(\text{exactly one}) = P(A \cap B^c) + P(A^c \cap B)$ We already have $P(A \cap B^c) = 0.1080$. Now let's compute $P(A^c \cap B)$: $P(A^c \cap B) = P(A^c) \times P(B) = 0.6 \times 0.73 = 0.4380$ Thus: $P(\text{exactly one}) = 0.1080 + 0.4380 = 0.5460$

Summary of Results:

• (a) $P(A \cap B) = 0.2920$
• (b) $P(A \cap B^c) = 0.1080$
• (c) Probability that neither $A$ nor $B$ occurs = $0.1620$
• (d) Probability that exactly one of $A$ or $B$ occurs = $0.5460$

Would you like more detailed explanations on any of these steps? Let me know if you have any further questions.

Here are 5 follow-up questions to consider:

1. How would the calculations change if $A$ and $B$ were not independent?
2. Can you explain how to derive the formula for $P(A \cap B)$ for dependent events?
3. How would you compute $P(A^c \cup B^c)$ instead of their intersection?
4. What is the relationship between $P(A \cup B)$ and $P(A \cap B)$?
5. How does the complement rule help in solving probability problems?

Tip: Always ensure you check whether events are independent before applying the multiplication rule for their intersection.