JEE Main 2023 · 24 January · morning · Single correct · +4 / −1
Let Ω be the sample space and A ⊆ Ω be an event.
Given below are two statements :
(S1) : If P(A) = 0, then A = φ
(S2) : If P(A) = 1, then A = Ω
Then :
- (a) both (S1) and (S2) are true
- (b) both (S1) and (S2) are false
- (c) only (S2) is true
- (d) only (S1) is true
Answer and solution
Answer (b) both (S1) and (S2) are false
Solution
Maths sir · 4 steps
Do not flip the rule: the empty set has probability 0, but probability 0 does not force the empty set.
- What is guaranteed. The axioms give P(φ) = 0 and P(Ω) = 1. Both statements ask for the converse, so one counterexample is enough to break each.
- Break S1. Pick a point at random from the interval [0, 1], so Ω = [0, 1] and P of a sub-interval is its length. Take A = {1/2}. A single point has length 0, so P(A) = 0, yet A is not empty.
- Break S2. Use the complement of the same event, B = Ω ∖ {1/2}. Then P(B) = 1 - 0 = 1, but B ≠ Ω because the point 1/2 is missing.
- So. Each statement has a counterexample, so both (S1) and (S2) are false, which is option (b).
The idea to keep. P(φ) = 0 and P(Ω) = 1 always hold, but not the other way round. Probability 0 means almost impossible, not impossible. One counterexample is enough to sink an if-then statement.
Answer from the NTA final answer key. Solution by PrepWiser.