A and B are independent, with P(A B) = 1/6, P(A' B') = 1/3 and P(A) > P(B). What is P(A)?

Question 74

HardOriginal question

$A$ and $B$ are independent, with $P(A \cap B) = \frac{1}{6}$ , $P(A' \cap B') = \frac{1}{3}$ and $P(A) > P(B)$ . What is $P(A)$ ?

\frac{1}{6}

Correct answerYour answer

\frac{1}{3}

Correct answerYour answer

\frac{1}{2}

Correct answerYour answer

\frac{5}{6}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

With $p = P(A)$ and $q = P(B)$ : $pq = \frac{1}{6}$ and $(1 - p)(1 - q) = 1 - (p + q) + pq = \frac{1}{3}$ , so $p + q = \frac{5}{6}$ .
$p$ and $q$ are the roots of $t^{2} - \frac{5}{6}t + \frac{1}{6} = 0$ , namely $\frac{1}{2}$ and $\frac{1}{3}$ ; the larger is $P(A) = \frac{1}{2}$ .

Worth noting

$\frac{1}{3}$ is the smaller root, $P(B)$ ; the question says $P(A) > P(B)$ .

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