Question

Let Ω be the sample space, and A be a subset of Ω. Given below are two statements:

S1​: If P(A)=0, then A=∅.
S2​: If A=∅, then P(A)=0.
(a) S1​ is true, S2​ is false.
(b) S1​ is false, S2​ is true.
(c) Both S1​ and S2​ are true.
(d) Both S1​ and S2​ are false.

Answer 1

S1 is false as events can have a probability of 0 without being empty in a continuous sample space; S2 is true since the probability of an empty event occurring is indeed 0. Hence, the correct statement is that S1 is false and S2 is true.

Step-by-step explanation:

The question is evaluating two statements about probability with respect to a sample space Ω and an event A which is a subset of Ω. Specifically, we are asked to assess the truth of the following:

• S1: If P(A)=0, then A=∅.
• S2: If A=∅, then P(A)=0.

Let's look at each statement individually:

1. S2 is straightforward to confirm as true since the probability of an empty set occurring is indeed 0; if there are no outcomes in event A, then it cannot possibly occur.
2. S1, however, is a bit more complex. While it is generally true that the probability of an empty set (∅) is 0, it is not correct to say that any event with a probability of 0 must be an empty set. An event can have a probability of 0 even if it is not physically impossible but just extremely unlikely in the context of a continuous sample space. This is a nuance in probability theory often encapsulated in the distinction between impossible events and events of measure zero.

Therefore, S1 is false, and S2 is true. The correct answer to the statement question is: (b) S1 is false, S2 is true.