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.

• A. \(0.55\)
• B. \(0.65\)
• C. \(0.35\)
• D. \(0.45\)

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: \[ 1-P(A\cup B\cup C) \]

Answer 1

The Correct Option is D

Step 1: Switch to easy numbers.
Imagine the class has $100$ students, so each percentage becomes a head count. Maths $= 40$, Physics $= 30$, Chemistry $= 20$.

Step 2: Write the overlaps as counts.
Maths and Physics $= 20$, Maths and Chemistry $= 10$, Physics and Chemistry $= 10$, and all three $= 5$.

Step 3: Find how many study at least one subject.
Add the singles, subtract the pair-overlaps, add back the triple overlap: $40 + 30 + 20 - 20 - 10 - 10 + 5 = 55$.

Step 4: Find how many study none.
Out of $100$ students, $55$ study at least one, so $100 - 55 = 45$ study none.

Step 5: Turn this into a probability.
Picking one student at random, the chance they study none is $\frac{45}{100} = 0.45$.

Step 6: Sanity check with a quick read.
Just over half the class takes a subject, so a little under half takes nothing, which matches $0.45$. \[ \boxed{0.45} \]