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If A and B are two events such that A \subset B and P(B) \neq 0, then which of the following is correct?
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If A and B are two events such that A \subset B and P(B) \neq 0, then which of the following is correct?

A. P(A \mid B)=P(B) / P(A)

B. P(A \mid B)<P(A)

C. P(A \mid B) \geq P(A)

D. None of these

Solution:

C. P(A \mid B) \geq P(A)

A and B are two events in which A \subset B and P(B) \neq 0 respectively.

As A \subset B \Rightarrow A \cap B=A

\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A})

As A \subset B \Rightarrow P(A)<P(B)

As we know \mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{\mathrm{P}(\mathrm{A})}{\mathrm{P}(\mathrm{B})} \neq \frac{\mathrm{P}(\mathrm{B})}{\mathrm{P}(\mathrm{A})} \ldots(1)

Consider

P(A \mid B)=\frac{P(A \cap B)}{P(B)}=\frac{P(A)}{P(B)}

(2)

It’s also common knowledge that \mathrm{P}(\mathrm{B}) \leq 1

\Rightarrow \frac{1}{P(B)} \geq 1

\Rightarrow \frac{\mathrm{P}(\mathrm{A})}{\mathrm{P}(\mathrm{B})} \geq \mathrm{P}(\mathrm{A})

\Rightarrow P(A \mid B) \geq P(A) \ldots

(3)

As a result, the right response is (C).

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