Question

If $E$ and $F$ are event 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 these

Answer 1

Answer: (D)

None of these

Consider the following events in single throw of a die.
$E=$ getting a multiple of $3;$
$F=$ getting an even number.
We have, $P\left(E\right)=\frac{1}{3}$ and $P\left(F\right)=\frac{1}{2}$.
Clearly, $P\left(E\right)\le P\left(F\right)$ and $P(E\cap F)=\frac{1}{6}>0$.
But none of the option (A),(B) and (C) is true.
Thus, the statement is alternatives (A), (B) and (C) do not necessarily hold good.