select the correct answer. which statement implies that a and b are independent events? a. (p(b|a)=p(bcap…

select the correct answer. which statement implies that a and b are independent events? a. (p(b|a)=p(bcap a)) b. (p(b|a)=\frac{p(b)}{p(a)}) c. (p(b|a)=p(a)) d. (p(b|a)=p(b))

select the correct answer. which statement implies that a and b are independent events? a. (p(b|a)=p(bcap a)) b. (p(b|a)=\frac{p(b)}{p(a)}) c. (p(b|a)=p(a)) d. (p(b|a)=p(b))

Answer

Answer:

D. $P(B|A) = P(B)$

Explanation:

Step1: Recall definition of independence

Two events $A$ and $B$ are independent if the occurrence of $A$ does not affect the probability of $B$.

Step2: Use conditional - probability formula

The formula for conditional probability is $P(B|A)=\frac{P(A\cap B)}{P(A)}$. For independent events, $P(A\cap B)=P(A)\times P(B)$. Substituting this into the conditional - probability formula gives $P(B|A)=\frac{P(A)\times P(B)}{P(A)} = P(B)$. So when $P(B|A) = P(B)$, events $A$ and $B$ are independent.