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 $P(E/F)\ge P(E)$ 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)

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

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