Question

which two statements about conditional probability are true? a. conditional probability, p(a), is the probability that event a will occur. b. conditional probability, p(a|b), is the probability that both event a and event b will occur. c. conditional probability, p(a|b), is the probability of event a occurring given that event b has occurred. d. the probability that it is raining outside is an example of conditional probability. e. the probability that it is both raining and cloudy outside is an example of conditional probability. f. the probability that it is raining if we already know it is cloudy outside is an example of conditional probability.

Explanation:

• Option A: \( P(A) \) is marginal probability, not conditional. So A is false.
• Option B: \( P(A \cap B) \) (joint probability) is for both A and B, not \( P(A|B) \). So B is false.
• Option C: By definition, \( P(A|B) \) is the probability of A given B has occurred. So C is true.
• Option D: The probability of rain alone is marginal, not conditional. So D is false.
• Option E: The probability of both rain and cloudy is joint probability, not conditional. So E is false.
• Option F: The probability of rain given it's cloudy (a known condition) fits the definition of conditional probability. So F is true.

Answer:

C. Conditional probability, \( P(A|B) \), is the probability of event \( A \) occurring given that event \( B \) has occurred.
F. The probability that it is raining if we already know it is cloudy outside is an example of conditional probability.