Question

Let $A$ and $B$ be two non-null events such that $A\subset B.$ Then, which of the following statements is always correct?

• A. $P\left(A|B\right)=1$
• B. $P\left(A|B\right)\le P\left(A\right)$
• C. $P\left(A|B\right)\ge P\left(A\right)$
• D. $P\left(A|B\right)=P\left(B\right)-P\left(A\right)$

Answer 1

The correct option is C $P\left(A|B\right)\ge P\left(A\right)$

Given,
$A\subset B\Rightarrow A\cap B=A$
$\Rightarrow P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}=\frac{P\left(A\right)}{P\left(B\right)}$
$\Rightarrow P\left(A/B\right)=\frac{P\left(A\right)}{P\left(B\right)}\ge P\left(A\right)$
$(\because 0\le P(B)\le 1)$