Question

if events a and b are independent, and the probability that event a occurs is 83%, what must be true? the probability that event b occurs is 17%. the probability that event b occurs is 83%. the probability that event a occurs, given that event b occurs, is 83%. the probability that event b occurs, given that event a occurs, is 83%.

Explanation:

Step1: Recall the definition of independent events

If A and B are independent events, then $P(A\cap B)=P(A)\times P(B)$ and $P(A|B) = P(A)$ and $P(B|A)=P(B)$. The probability of event A occurring does not depend on whether event B occurs and vice - versa.

Step2: Analyze each option

• Option 1: Just because $P(A) = 83\%$, there is no reason for $P(B)$ to be $17\%$.
• Option 2: There is no reason for $P(B)$ to be equal to $P(A)$ just because A and B are independent.
• Option 3: Since A and B are independent, by the definition of independent events, the conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{P(A)\times P(B)}{P(B)} = P(A)$. Given $P(A)=83\%$, so $P(A|B) = 83\%$.
• Option 4: Since A and B are independent, $P(B|A)=P(B)$, and we have no information to suggest that $P(B)=83\%$.

Answer:

The probability that event A occurs, given that event B occurs, is 83%. So the correct option is: The probability that event A occurs, given that event B occurs, is 83%.