For events A and B with P(A) > 0 and P(B) > 0, which of the following statements are true? I. If A B, then P(B A) = 1. II. If A B, then P(A B) = (P(A))/(P(B)). III. If A and B are mutually exclusive, then P(A B) = 0. IV. P(A B) ≥ P(A) for any two such events.

Question 22

MediumOriginal question

For events $A$ and $B$ with $P(A) > 0$ and $P(B) > 0$ , which of the following statements are true?
I. If $A \subset B$ , then $P(B \mid A) = 1$ .
II. If $A \subset B$ , then $P(A \mid B) = \frac{P(A)}{P(B)}$ .
III. If $A$ and $B$ are mutually exclusive, then $P(A \mid B) = 0$ .
IV. $P(A \mid B) \ge P(A)$ for any two such events.

I, II and III onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerI and II onlyCorrect answerYour answerI and III onlyCorrect answerYour answer

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CorrectNot quiteAnswer A

Correct answer: A

I: $A \cap B = A$ , so $P(B \mid A) = \frac{P(A)}{P(A)} = 1$ . II: $P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{P(A)}{P(B)}$ . III: $P(A \cap B) = 0$ , so $P(A \mid B) = 0$ .
IV is false: for mutually exclusive events $P(A \mid B) = 0$ , which is less than $P(A)$ .

Worth noting

A condition can lower a probability as well as raise it.

Concept: Properties of conditional probability
Typical timeTypical: 120 s
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The rule this testsConditional probability

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