Question

A multiple choice test question has five altemative answers, of which only one is correct. If a student has done his home work, then he is sure to identify the correct answer; otherwise, he chooses an answer at random.
Let $ E$ : denotes the event that a student does his home work with $P ( E )= p$
and $F$ : denotes the event that he answer the question correctly.
The relation $P ( E / F ) \geq P ( E )$ holds good for

• A. all values of $p$ in $[0,1]$
• B. all values of $p$ in $(0,1)$ only
• C. all values of $p$ in $[0.5,1]$ only
• D. no value of $p$.

Answer 1

Answer: (A)

$P(E)=p$
$P(F) =P(E \cap F)+P(\bar{E} \cap F) $
$P(F) =P(E) P(F / E)+P(\bar{E}) P(F / E) $
$ =p \cdot 1+(1-p) \cdot \frac{1}{5}=\frac{4 p}{5}+\frac{1}{5}=\frac{4 p+1}{5}$

now $ P(E / F)=\frac{5 p}{(4 p+1)} \geq p$
equality holds for $p =0$ or $p =1$
for all others value of $p \in(0,1), $ LHS $>$ RHS, hence (A)