Conditional probability on a multiple choice test

conditional probabilityprobability

A student is taking a multiple-choice test. Each question on the test has five possible answers and only one choice is correct. The probability this student knows the correct answer is 70%. If the student does not know the answer, they select an answer at random with each answer having an equal probability of being picked. Calculate the conditional probability that the student knew the answer given they answered the question correctly.

I started off like this:
Let B denote the event that the student knew the answer. Let A denote the event that the student answered the question correctly.

I was able to work out that $P(A)=\frac{3}{10}*\frac{1}{5} + \frac{7}{10} = \frac{19}{25}$

And I know the formula $P(B|A)=\frac{P(A \cap B)}{P(A)}$

I am unsure on how to work out $P(A \cap B)$. Any advice would be greatly apprectiated.

Best Answer

It is easiest to draw a tree for these.


know        70%   -   correct   1       .7

                  /   correct   1/5     .06
don't know  30%   
                  \   incorrect 4/5     .24

Thus $P(A\cap B)=.7$ and your desired probability is $\frac{P(A\cap B)}{P(A)}=\frac{.7}{.76}= \frac{35}{38}$. It is very likely that the student knew the answer if they got it correct.

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