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 events A and B are independent, 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 or not, and vice - versa. Given $P(A) = 0.83$.

Step2: Analyze each option

• Option 1: There is no relation between $P(A)$ and $P(B)$ that says $P(B)=1 - P(A)$ for independent events.
• 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 conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{P(A)\times P(B)}{P(B)} = P(A)$. Since $P(A)=0.83$, $P(A|B) = 0.83$ or 83%.
• Option 4: $P(B|A)=P(B)$, and there is no information to suggest $P(B)=0.83$.

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%.