Question

The probability that a student knows the correct answer to a multiple choice question is 2/3. If the student does not know the answer, then the student guesses the answer. The probability of the gussed answer being correct is 1/4. Given that the student has answreed the question correctly, the conditional probability that the student knows the correct answer is

• A. $\frac{2}{3}$
• B. $\frac{3}{4}$
• C. $\frac{5}{6}$
• D. $\frac{8}{9}$

Answer 1

The correct option is D $\frac{8}{9}$

Let $E_1$ be the student who knows the answer. $E_2$ be the student who guesses the answer. A be the event for correct answer.

$P\left(E_1\right)=\frac{2}{3}$

$P\left(E_2\right)=\frac{1}{3}$

$P\left(A/E_2\right)=\frac{1}{4}$

$P\left(A/E_1\right)=1$

By Baye's theorem,
$P\left(E_1/A\right)=\frac{P\left(E_1\right)P\left(A/E_1\right)}{P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)}$

$=\frac{\frac{2}{3}\times 1}{\frac{2}{3}\times 1+\frac{1}{3}\times \frac{1}{4}}=\frac{8}{9}$