Question

If $E$ and $F$ are events with $P\left(E\right)\le P\left(F\right)$ and $P(E\cap F)>0$, then

• A. Occurrence of $E$ $\Rightarrow$ Occurrence of $F$
• B. Occurrence of $F$ $\Rightarrow$ Occurrence of $E$
• C. Non-occurrence of $E$ $\Rightarrow$ Non-occurrence of $F$
• D. None of the above implications holds

Answer 1

The correct option is D

None of the above implications holds

Explanation for the correct option:

Given that $P\left(E\right)\le P\left(F\right)$

$\Rightarrow$ $n\left(E\right)\le n\left(F\right)$

$P(E\cap F)>0$

$\Rightarrow$$E\cap F\ne \phi$

These two above mentioned conditions does not mean that $E$ is a subset of $F$ or $F$ is a subset of $E$

That is,

$\begin{array}{rcl}E& \subseteq & F\end{array}$ or, $\begin{array}{rcl}F& \subseteq & E\end{array}$ or, $\begin{array}{rcl}E’& \subseteq & F’\end{array}$

Hence all the three mentioned implications are wrong.

Hence, option D is correct