Question

the probability for event a is 0.4, the probability for event b is 0.2, and the probability of events a and b is 0.1. why are the events not independent? the sum of p(a) and p(b) is greater than p(a and b). the product of p(a) and p(b) is greater than p(a and b). the product of p(a) and p(b) is not equal to p(a and b). the sum of p(a) and p(b) is not equal to p(a and b).

Explanation:

To determine if two events \( A \) and \( B \) are independent, we use the rule that for independent events, \( P(A \text{ and } B)=P(A)\times P(B) \).

1. Calculate \( P(A)\times P(B) \): Given \( P(A) = 0.4 \) and \( P(B)=0.2 \), we have \( P(A)\times P(B)=0.4\times0.2 = 0.08 \).
2. Compare with \( P(A \text{ and } B) \): We know \( P(A \text{ and } B) = 0.1 \). Since \( 0.08

eq0.1 \), the product of \( P(A) \) and \( P(B) \) is not equal to \( P(A \text{ and } B) \), which means the events are not independent.

The other options are incorrect: the sum of probabilities is not related to the independence rule, and the key for independence is the product of individual probabilities equaling the joint probability, not the sum.

Answer:

C. The product of \( P(A) \) and \( P(B) \) is not equal to \( P(A \text{ and } B) \)