If A, B and C are three independent events such that P(A) = P(B) = P(C) = \( x \), then P(at least two of A, B, C occur) =
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 3x^2-2x^3 \)
“At least two” means exactly two or all three. By independence:
\( \displaystyle P(\text{exactly two})=3\cdot x\cdot x\cdot(1-x) \)
\( \displaystyle =3x^2-3x^3 \)
\( \displaystyle P(\text{all three})=x^3 \)
\( \displaystyle P(\text{at least two})=3x^2-3x^3+x^3 \)
\( \displaystyle =3x^2-2x^3 \)
Shortcut: Put x = 1: then all three events are certain, so the answer must be 1. The options give 7, −1, 5 and 1, and only Option (4) gives 1.
NCERT Exemplar:
Probability
