https://file.aiquickdraw.com/mathbot/file/1728965729890dhg3p39e.jpg

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.

A random experiment produces many possible events. Two of these events are event A and event B. These events are independent of each other and occur with P(A) = 0.4 and P(B) = 0.73. Compute the following probabilities: (a) P(A ∩ B), (b) P(A ∩ B^c), (c) Probability that neither A nor B occurs, (d) Probability that exactly one of these two events will occur.

Learn how to calculate probabilities involving independent events A and B with given probabilities P(A) = 0.4 and P(B) = 0.73. The solution covers key probability rules, including intersection, complement, and the probability of neither or exactly one event occurring.

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