Question

A multiple-choice question has $5$ options of which only one is correct. If a student does home work, then he is sure to identify the correct answer, otherwise the answer is chosen at random. Let $A$ be the event that the student does his home work and $B$ be the event that the student answers correctly. If $P\left(A\right)=\frac{2}{3}$ , then $P\left(\frac{A}{B}\right)$ is equal to

• A. $\frac{10}{11}$
• B. $\frac{4}{5}$
• C. $\frac{3}{7}$
• D. $\frac{6}{7}$

Answer 1

Answer: (A)

$P\left(\frac{B}{A^{c}}\right)=\frac{1}{5}, P(A)=\frac{2}{3}$
$P(A / B)=\frac{P(A \cap B)}{P(B)}=\frac{P(A) \cdot P(B / A)}{P(A) P\left(\frac{B}{A}\right)+P\left(A^{c}\right) P\left(B / A^{c}\right)}$
$=\frac{\frac{2}{3} \times 1}{\frac{2}{3} \times 1+\frac{1}{3} \times \frac{1}{5}}=\frac{10}{11}$