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 the marginal probability of event \( A \), not conditional probability. So A is false.
• Option B: The probability that both \( A \) and \( B \) occur is the joint probability \( P(A\cap B) \), not \( P(A|B) \). So B is false.
• Option C: By definition, \( P(A|B) \) is the probability of \( A \) given \( B \) has occurred. This is the correct definition of conditional probability. So C is true.
• Option D: The probability that it is raining outside is a marginal probability (no condition given), not conditional. So D is false.
• Option E: The probability that it is both raining and cloudy is a joint probability (\( P(\text{raining} \cap \text{cloudy}) \)), not conditional. So E is false.
• Option F: The probability of raining given that it is cloudy (a condition) is a classic example of conditional probability (\( P(\text{raining}|\text{cloudy}) \)). 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.