Question

In an entrance test, there are multiple choice questions. There are four possible options of which one is correct. The probability that a student knows the answer to a question is 90%. If he gets the correct answer to a question, then the probability that he was guessing is
A. $\dfrac{1}{{37}}$
B. $\dfrac{{36}}{{37}}$
C. $\dfrac{1}{4}$
D. $\dfrac{1}{{49}}$

Answer 1

Hint: According to the given in the question we have to determine the probability that he was guessing when in an entrance test, there are multiple choice questions. There are four possible options of which one is correct. The probability that a student knows the answer to a question is 90%. if he gets the correct answer to a question. So, First of all we have to understand the probability which is an explained below:
Probability: The probability provides the ratio of the number of the favourable outcomes to the total number of possible outcomes.
We have to define the events ${A_1}$ and ${A_2}$ where ${A_1}$ is the event when he knows the answer and ${A_2}$ is the event when he does not know the answer and E is the event when he gets the correct answer. Now, we have to determine the probabilities $P({A_1})$ and $P({A_2})$. As we know that if the probability of the event to occur is X, then the probability of the event not occurring is 1-X. So we can determine $P({A_2})$ with the help of the formula as mentioned below:

Formula used:
$ \Rightarrow P({A_2}) = 1 - P({A_1})$……………………..(1)
Now to determine the required probability we have to use the formula as mentioned below:
$ \Rightarrow P\left( {\dfrac{{{A_2}}}{E}} \right) = \dfrac{{P({A_2})P\left( {\dfrac{E}{{{A_2}}}} \right)}}{{P({A_1})P\left( {\dfrac{E}{{{A_1}}}} \right) + P({A_2})P\left( {\dfrac{E}{{{A_2}}}} \right)}}$…………………………………(2)
Now, on substituting all the values in the formula (2) as mentioned above, we can determine the required probability.

Complete step by step answer:
Step 1: First of all we have to determine the probability $P({A_1})$which is the probability when he knows the answer. Hence,
$ \Rightarrow P({A_1}) = \dfrac{9}{{10}}$
Step 2: Now, we have to determine the probability $P({A_2})$ which is the probability that he does not know the answer. We can determine it with the help of the formula (1) as mentioned in the solution hint. Hence,
$
   \Rightarrow P({A_2}) = 1 - \dfrac{9}{{10}} \\
   \Rightarrow P({A_2}) = \dfrac{1}{{10}} \\
 $
Step 3: Now, as we have obtained the probability $P({A_1})$ and $P({A_2})$ from the step 1 and step 2 and to determine the required probability we have to substitute these probability in the formula (2) as mentioned in the solution hint.
$ \Rightarrow P\left( {\dfrac{{{A_2}}}{E}} \right) = \dfrac{{\dfrac{1}{{10}}.\dfrac{1}{4}}}{{\dfrac{9}{{10}}.1 + \dfrac{1}{{10}}.\dfrac{1}{4}}}$
Now, to solve the expression above we have to find the LCM
$
   \Rightarrow P\left( {\dfrac{{{A_2}}}{E}} \right) = \dfrac{{\dfrac{1}{{40}}}}{{\dfrac{9}{{10}} + \dfrac{1}{{40}}}} \\
   \Rightarrow P\left( {\dfrac{{{A_2}}}{E}} \right) = \dfrac{{\dfrac{1}{{40}}}}{{\dfrac{{36 + 1}}{{40}}}} \\
   \Rightarrow P\left( {\dfrac{{{A_2}}}{E}} \right) = \dfrac{1}{{37}} \\
 $

Hence, we have obtained the probability that he was guessing the answer of the question is $\dfrac{1}{{37}}$. Therefore, option (A) is correct.

Note: To determine the probability we have to divide the favourable event by the sample space or total number of outcomes.
If the probability of an event to occur is P(A) then the probability of an event that does not occur is 1-P(A).