Probability 3 Question 3
3. Let $A$ and $B$ be two non-null events such that $A \subset B$. Then, which of the following statements is always correct.
(2019 Main, 8 April I)
(a) $P(A / B)=P(B)-P(A)$
(b) $P(A / B) \geq P(A)$
(c) $P(A / B) \leq P(A)$
(d) $P(A / B)=1$
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Solution:
- We know that, $P(A / B)=\frac{P(A \cap B)}{P(B)}$
[by the definition of conditional probability]
$$ \begin{array}{lc} \because & A \subset B \\ \Rightarrow & A \cap B=A \\ \therefore & P(A / B)=\frac{P(A)}{P(B)} \end{array} $$
As we know that, $0 \leq P(B) \leq 1$
$$ \begin{array}{lll} \therefore & 1 & \leq \frac{1}{P(B)}<\infty \Rightarrow P(A) \leq \frac{P(A)}{P(B)}<\infty \\ \Rightarrow & \frac{P(A)}{P(B)} & \geq P(A) \end{array} $$
Now, from Eqs (i) and (ii), we get
$$ P(A / B) \geq P(A) $$
