Let's go through each question one by one, using the given information:

• P(only A) = 0.48
• P(only B) = 0.09
• P(A ∪ B) = 0.84

a) What is P(A)?

The total probability of event A occurring is given by:

$P(A) = P(\text{only A}) + P(\text{A and B occurring})$

From the given data:

• P(only A) = 0.48
• We don’t know P(A and B occurring) yet, but we can find it from the probability of the union of A and B:

The union of events A and B is calculated as:

$P(A \cup B) = P(\text{only A}) + P(\text{only B}) + P(\text{A and B occurring})$

Substituting the given values:

$0.84 = 0.48 + 0.09 + P(\text{A and B occurring})$

$P(\text{A and B occurring}) = 0.84 - 0.48 - 0.09 = 0.27$

Now, we can calculate P(A):

$P(A) = 0.48 + 0.27 = 0.75$

So, the answer is B. 0.75.

b) What is P(neither A nor B)?

The probability of neither event A nor event B occurring is the complement of the union of A and B. Thus:

$P(\text{neither A nor B}) = 1 - P(A \cup B)$

Substituting the given value:

$P(\text{neither A nor B}) = 1 - 0.84 = 0.16$

So, the answer is E. 0.16.

c) What is P(only one of the two events (A or B) occurring)?

The probability of only one of the events (A or B) occurring can be found by adding the probabilities of only A and only B:

$P(\text{only one of A or B}) = P(\text{only A}) + P(\text{only B})$

Substituting the given values:

$P(\text{only one of A or B}) = 0.48 + 0.09 = 0.57$

So, the answer is B. 0.57.

d) Are A and B independent events?

Two events A and B are independent if the probability of both events occurring together is the product of their individual probabilities:

$P(A \cap B) = P(A) \times P(B)$

We already know P(A ∩ B) = 0.27, P(A) = 0.75, and P(B) = P(\text{only B}) + P(\text{A and B}) = 0.09 + 0.27 = 0.36.

Now, check if they satisfy the condition for independence:

$P(A \cap B) = 0.27 \quad \text{and} \quad P(A) \times P(B) = 0.75 \times 0.36 = 0.27$

Since these are equal, A and B are independent.

So, the answer is B. Yes.

e) Are A and B mutually exclusive events?

Two events A and B are mutually exclusive if they cannot occur together, i.e., P(A ∩ B) = 0.

We know that P(A ∩ B) = 0.27, which is greater than zero, so the events are not mutually exclusive.

So, the answer is A. No.

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Summary of answers:

• a) B. 0.75
• b) E. 0.16
• c) B. 0.57
• d) B. Yes
• e) A. No

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Would you like further clarification or more examples? Here are some related questions to explore further:

1. How would you calculate P(A ∩ B) if the union and individual probabilities were different?
2. What happens if P(A ∩ B) = 0, how would this affect the independence and exclusivity?
3. How can you verify if events are mutually exclusive using a Venn diagram?
4. If P(A ∪ B) = 0.70, and P(only A) = 0.40, what would be the probability of only B?
5. How does P(A ∩ B) change if A and B are not independent?

Tip: Always remember to use the formula for the union of two events when given partial probabilities:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$