Let $\Omega$ be the sample space and $\mathrm{A} \subseteq \Omega$ be an event. Given below are two statements: (S1): If $\mathrm{P}(\mathrm{A})=0$, then $\mathrm{A}=\phi$ (S2): If $\mathrm{P}(\mathrm{A})=1$, then $\mathrm{A}=\Omega$ Then

(D) both (S1) and (S2) are false

(C) both (S1) and (S2) are true

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Probability

Let

Ω

be the sample space and

A \subseteq 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

Answer: (D)

Both claims fail once the sample space is continuous. With

Ω = [0, 1]

under the uniform law and

A = {\frac{1}{2}}

, a single point has probability

0

while

A \neq = ϕ

, so

P (A) = 0

need not give

A = ϕ

. With

A = [0, 1)

the probability is

1

because the omitted point carries no measure, yet

A \neq = Ω

, so

P (A) = 1

need not give

A = Ω

. Both statements are therefore false.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Single correct option