Question

In a class of 125 students 70 passed in Mathematics, 55 in Statistics and 30 in both. The probability that a student selected at random from the class, has passed in only one subject is
A.\[\dfrac{{13}}{{25}}\]
B.\[\dfrac{3}{{25}}\]
C.\[\dfrac{{17}}{{25}}\]
D.\[\dfrac{8}{{25}}\]

Answer 1

Hint: First we will first use the formula of the probability of any event happening is given by dividing the number of outcomes of that event divided by the total number of events, that is; $P = \dfrac{{{\text{Number of outcomes}}}}{{{\text{Total number of outcomes}}}}$to find the values of \[P\left( M \right)\], \[P\left( S \right)\] and \[P\left( {M \cap S} \right)\]. Then we will use these values to substitute in \[P\left( {M \cup S} \right) = P\left( M \right) + P\left( S \right) - P\left( {M \cap S} \right)\] to find the required value.

Complete step-by-step answer:
We are given that there are a total 125 students.
We are also given that there are 70 students passed in mathematics, 55 in statistics and 30 in both.
We know that the probability of any event happening is given by dividing the number of outcomes of that event divided by the total number of events, that is; $P = \dfrac{{{\text{Number of outcomes}}}}{{{\text{Total number of outcomes}}}}$.
Finding the probability of students passed in mathematics from the above formula of probability, we get
\[ \Rightarrow P\left( M \right) = \dfrac{{70}}{{125}}\]
Computing the probability of students passed in statistics from the above formula of probability, we get
\[ \Rightarrow P\left( S \right) = \dfrac{{55}}{{125}}\]
Finding the probability of students passed in both mathematics and statistics from the above formula of probability, we get
\[ \Rightarrow P\left( {M \cap S} \right) = \dfrac{{30}}{{125}}\]
Let us assume that \[P\left( {M \cup S} \right)\] represents the probability of students passed in any of the subjects.
Using the formula \[P\left( {M \cup S} \right) = P\left( M \right) + P\left( S \right) - P\left( {M \cap S} \right)\] and the above values, we get
\[
   \Rightarrow P\left( {M \cup S} \right) = \dfrac{{70}}{{125}} + \dfrac{{55}}{{125}} - \dfrac{{30}}{{125}} \\
   \Rightarrow P\left( {M \cup S} \right) = \dfrac{{70 + 55 - 30}}{{125}} \\
   \Rightarrow P\left( {M \cup S} \right) = \dfrac{{95}}{{125}} \\
   \Rightarrow P\left( {M \cup S} \right) = \dfrac{{13}}{{25}} \\
 \]
Hence, option A is correct.

Note: In solving these types of questions, you should be familiar with the formula to find the probability of any event happening and not happening. Some students get confused while applying the formulae. In this question, one can find the probability for both subjects instead of one of the subjects and then conclude the wrong answer.