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 cap 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, not conditional, probability.
• Option B: \( P(A \cap B) \) (joint probability) is not conditional probability.
• Option C: Matches the definition of \( P(A|B) \) (conditional probability).
• Option D: “Raining” alone is marginal probability, not conditional.
• Option E: “Raining and cloudy” is joint probability, not conditional.
• Option F: “Raining given cloudy” fits the conditional probability definition (probability of an event given another event has occurred).

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.