Question

Suppose two events A and B are mutually exclusive, with P(A) ≠ 0 and P(B) ≠ 0. By working through the following steps, you'll see why two mutually exclusive events are not independent. (a) For mutually exclusive events, can event A occur if event B has occurred? Yes. By definition, mutually exclusive events can occur together. No. Two events will never occur concurrently. Yes. Any two events can occur concurrently. No. By definition, mutually exclusive events cannot occur together. What is the value of P(A|B)? (b) Using the information from part (a), can you conclude that events A and B are not independent if they are mutually exclusive? Explain. Yes. Because P(A|B) ≠ P(A), the events A and B are not independent. No. Two events being mutually exclusive always implies independence. No. Because P(A|B) ≠ P(A), the events A and B may be independent. Yes. Two events being mutually exclusive always implies independence.

Answer 1

Answer with Step-by-step explanation:

We are given that two events A and B are mutually exclusive.

$A\cap B=\phi$

$P(A\cap B)=0$

$P(A)\neq 0$

$P(B)\neq 0$

a.For mutually exclusive events,

$P(A/B)=\frac{P(A\cap B)}{P(B)}=\frac{0}{P(A)}=0$

Therefore, event A can not occurred if event B has occurred because two events can not occur together.

Answer:No, by definition mutually exclusive events cannot occur together.

b.When two events are independent

Then , $P(A\cap B)=P(A)\cdot P(B)$

$P(A/B)=\frac{P(A)\cdot P(B)}{P(B)}=P(A)$

If two events are mutually exclusive then

$P(A\cap B)=0$

Then , $P(A/B)=0$

Therefore, $P(A/B)\neq P(A)$

Hence, we can concluded that events A and B are not independent if they are mutually exclusive.

Answer:Yes, $P(A/B)\neq P(A)$