[Math] Probability that a student knows the answer

probability

The Probality 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 probality of the guessed answer being correct is 1/4 . Given that the student has answered the questions correctly , the conditional probability that the student knows the correct answer is _

I started solving it . But I'm stuck .

Let

a = student knows the correct answer

b = guessed answer is correct

P(a)=2/3 P(a')=1/3

P(b)=1/4 P(b')=3/4

Let c = answered all questions correctly

I guess

P(c)=P(a)+P(a').P(b)

P(c)=(2/3) +((1/3).(1/4))

P(c) = 3/4

TO find

P(a/c)=P(a n c)/p(c)

I dont know how to find p(a n c) . Please elaborate and Help !!!

Best Answer

Let's define the events as:

\begin{eqnarray*} A &=& \mbox{Student knows the correct answer} \\ C &=& \mbox{Student answered correctly.} \\ \end{eqnarray*}

You are asked to find a conditional probability, $P(A \mid C)$. We can use Bayes' Theorem to calculate it:

\begin{eqnarray*} P(A \mid C) &=& \dfrac{P(C \mid A)P(A)}{P(C \mid A)P(A) + P(C \mid A^c)P(A^c)} \\ && \\ &=& \dfrac{1 \times \frac{2}{3}}{1 \times \frac{2}{3} + \frac{1}{4} \times \frac{1}{3}} \\ && \\ &=& \dfrac{8}{9} \end{eqnarray*}

Note:

  1. $P(C \mid A) = 1$ because, given that the student knows the answer, the answer must be correct.

  2. Event $A^c$ is "the student guessed" because he guesses if he doesn't know the answer.

Related Question