$P(A) = 0.4$ , $P(B) = 0.5$ and $P(A \cup B) = 0.7$ . Which of the following statements are true?
I. $A$ and $B$ are independent.
II. $A$ and $B$ are mutually exclusive.
III. $P(A' \cap B) = 0.3$ .
IV. $P(A \mid B) = 0.5$ .

I and III onlyCorrect answerYour answerI, III and IV onlyCorrect answerYour answerII and III onlyCorrect answerYour answerIII and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$P(A \cap B) = 0.4 + 0.5 - 0.7 = 0.2 = 0.4 \times 0.5$ : I is true, and II is false since $P(A \cap B) \ne 0$ .
III: $0.5 - 0.2 = 0.3$ , true. IV: $\frac{0.2}{0.5} = 0.4$ , false.

Worth noting

IV takes $P(B)$ for $P(A \mid B)$ ; for independent events $P(A \mid B) = P(A) = 0.4$ .

Concept: Independence and exclusiveness
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsIndependent events

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Ten questions from Conditional Probability & Bayes' Theorem, at your own pace.

Explanation