Question

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let \[\dfrac{3}{4}\]be the probability that he knows the answer and \[\dfrac{1}{4}\]be the probability that he guesses. Assuming that a student who guesses the answer will be correct with probability \[\dfrac{1}{4}\]. What is the probability that he knows the answer given that he answered it correctly?

Answer 1

Hint: This problem is solved based on conditional probability theorem it states that likelihood of an event or outcome occurring, based on the occurrence of previous event or outcome and formula for the above problem is as follows \[P\left( \dfrac{A}{C} \right)=\dfrac{P(A)P\left( \dfrac{C}{A}\right)} {P(A)P\left( \dfrac{C}{A} \right)+P(B)P\left( \dfrac{C}{B} \right)}\] and each terms in formula are specified below
Let A, B, C be three events as follows
A: student knows the answer
B: student guesses
C: student answers correctly

Complete step-by-step solution -
We need to find the probability that the student knows the answer, if he answered it correctly
\[P\left( \dfrac{A}{C} \right)=\dfrac{P(A)P\left( \dfrac{C}{A} \right)}{P(A)P\left( \dfrac{C}{A} \right)+P(B)P\left( \dfrac{C}{B} \right)}\]. . . . . . . . . . . . . . . . . . . . (1)
Let us now specify the each term in the above formulae
\[P(A)\]=probability that student knows the answer =\[\dfrac{3}{4}\]
\[P(B)\]=probability that student guesses the answer=\[\dfrac{1}{4}\]
\[P\left( \dfrac{C}{A} \right)\]=probability that student answers correctly, if he knows the answer=\[1\]
\[P\left( \dfrac{C}{B} \right)\]=probability that student answers correctly, if he guesses=\[\dfrac{1}{4}\]
Now substitute the values in the above formula (1), we will get
\[P\left( \dfrac{A}{C} \right)=\dfrac{\dfrac{3}{4}\times 1}{\dfrac{1}{4}\times \dfrac{1}{4}+\dfrac{3}{4}\times 1}\]
\[P\left( \dfrac{A}{C} \right)=\dfrac{\dfrac{3}{4}}{\dfrac{13}{16}}\]
\[P\left( \dfrac{A}{C} \right)=\dfrac{12}{13}\]
Therefore, probability that the student knows the answer given that he answered it correctly is \[\dfrac{12}{13}\]

Note: Note that conditional probability states that probability of an event occurring given that another event has already occurred but it didn’t state that both the two events can occur simultaneously. The conditional probability theorem is applied to events which are neither independent nor mutually exclusive.