Probability Theory – True Statements When P(A) = 0

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Let A and B be events in the sample space, where $P(A) = 0$. Which of
the following must be TRUE?

(a) A and B are independent.

(b) A and B are mutually exclusive.

(c) A must be a subset of B.

(d) None of the given options.

I was asked this question in one of my worksheets and the question stated to only choose one answer. Here is my analysis:

Since we know that $P(A) = 0$, then $P(A∩B) = 0$.

Thus, (b) is right and since for A and B to be both independent, $P(A∩B) = P(A)P(B) = 0$. And (a) is also right.

For option (c), since $P(A) = 0$, we implicitly know that A will be a subset of B since an empty set is a subset of any set. Thus (c) is also right?

In summary, I think that all (a), (b) and (c) are right options so should the answer be (d) in this case? Or could there be something wrong with this question?

Thanks for the help!!

Best Answer

Let a loaded dice be such that the only possible outcome is 6. Let $A=\{1, 2\}$ and $B = \{2, 3, 6\}$, then

  • $P(A) = 0$
  • $A\cap B = \{2\}\not = \emptyset$ hence $A$ and $B$ are not mutually exclusive
  • $A$ is not a subset of $B$

The only true sentence is (a) because if $P(A)=0$ then $0\le P(A\cap B)\le P(A) = 0 = P(A)P(B)$.

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