Question:
In a class XII of a school, \(40\%\) of the students study Mathematics, \(30\%\) study Physics and \(20\%\) study Chemistry. \(20\%\) of the class study both Mathematics and Physics, \(10\%\) study both Mathematics and Chemistry and \(10\%\) study both Physics and Chemistry. \(5\%\) of the class study all the three subjects. If a student is selected at random from the class, find the probability that he studies neither Mathematics nor Physics nor Chemistry.
In a class XII of a school, \(40\%\) of the students study Mathematics, \(30\%\) study Physics and \(20\%\) study Chemistry. \(20\%\) of the class study both Mathematics and Physics, \(10\%\) study both Mathematics and Chemistry and \(10\%\) study both Physics and Chemistry. \(5\%\) of the class study all the three subjects. If a student is selected at random from the class, find the probability that he studies neither Mathematics nor Physics nor Chemistry.
Show Hint
For three sets:
\[
P(A\cup B\cup C)
=
P(A)+P(B)+P(C)
-P(A\cap B)
-P(B\cap C)
-P(C\cap A)
+P(A\cap B\cap C)
\]
Probability of none:
\[
1-P(A\cup B\cup C)
\]
Updated On: Jun 16, 2026
- \(0.55\)
- \(0.65\)
- \(0.35\)
- \(0.45\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Concept:
Use the Principle of Inclusion-Exclusion:
\[\begin{aligned}
n(M\cup P\cup C)
&=n(M)+n(P)+n(C)\\
&\quad-n(M\cap P)-n(M\cap C)-n(P\cap C)\\
&\quad+n(M\cap P\cap C)
\end{aligned}\]
Step 1: Substitute the given percentages. \[\begin{aligned} P(M\cup P\cup C) &=0.40+0.30+0.20\\ &\quad-0.20-0.10-0.10\\ &\quad+0.05 \end{aligned}\] \[\begin{aligned} &=0.55 \end{aligned}\]
Step 2: Find the probability of studying none of the subjects. \[\begin{aligned} P(\text{None}) &=1-P(M\cup P\cup C) \end{aligned}\] \[\begin{aligned} &=1-0.55 \end{aligned}\] \[\begin{aligned} &=0.45 \end{aligned}\] \[\begin{aligned} \boxed{0.45} \end{aligned}\] Hence, option \(\mathbf{(D)}\) is correct.
Step 1: Substitute the given percentages. \[\begin{aligned} P(M\cup P\cup C) &=0.40+0.30+0.20\\ &\quad-0.20-0.10-0.10\\ &\quad+0.05 \end{aligned}\] \[\begin{aligned} &=0.55 \end{aligned}\]
Step 2: Find the probability of studying none of the subjects. \[\begin{aligned} P(\text{None}) &=1-P(M\cup P\cup C) \end{aligned}\] \[\begin{aligned} &=1-0.55 \end{aligned}\] \[\begin{aligned} &=0.45 \end{aligned}\] \[\begin{aligned} \boxed{0.45} \end{aligned}\] Hence, option \(\mathbf{(D)}\) is correct.
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