Probability: Question 5

Worked solution

Step 1: Test whether $A$ and $B$ are independent

Two events are independent exactly when $P(A \cap B) = P(A) \times P(B)$. Here: $P(A) \times P(B) = 0.4 \times 0.5 = 0.2$

Since this equals the given $P(A \cap B) = 0.2$, the events are independent.

Step 2: Test whether $A$ and $B$ are mutually exclusive

Two events are mutually exclusive exactly when $P(A \cap B) = 0$. Here $P(A \cap B) = 0.2 \neq 0$, so the events are not mutually exclusive, they can, and sometimes do, happen together.

Step 3: Combine the two conclusions

$A$ and $B$ are independent but not mutually exclusive, which matches option A.

Check by another method: if the events were independent, the conditional probability $P(A \mid B)$ should equal the unconditional $P(A)$: $P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.5} = 0.4 = P(A) \checkmark$ This confirms independence directly, agreeing with Step 1.

Why the other options are wrong

• B: this is false, since $P(A \cap B) = 0.2 \neq 0$ means the events are not mutually exclusive.
• C: two events with non-zero probabilities can never be both mutually exclusive and independent. If they were mutually exclusive, $P(A \cap B)$ would have to be $0$, but independence would then require $P(A)\times P(B)=0$, which is false here since both $P(A)=0.4$ and $P(B)=0.5$ are non-zero.
• D: this is false, since Step 1 shows the events are independent.

Final answer

• $A$ and $B$ are $\boxed{\text{independent, but not mutually exclusive}}$, option A.