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A student answers a multiple choice question with 5 alternatives, of which exactly one is correct. The probability that he knows the correct answer is $p, 0 < p < 1$. If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, the probability that he did not tick the answer randomly, is
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Verified Answer
The correct answer is:
$\frac{5 p}{4 p+1}$
Let $E_{1}=$ Student does not know the answer
$E_{2}=$ Student knows the answer
and $E=$ Student answer correctly
$\therefore \quad P\left(E_{1}\right)=1-p \Rightarrow P\left(E_{2}\right)=p$
$\Rightarrow \quad P\left(\frac{E}{E_{2}}\right)=1$ and $P\left(\frac{E}{E_{1}}\right)=\frac{1}{5}$
Note that, the probability that student did not know the answer randomly =The probability that student know the answer.
$$
\therefore \quad P\left(\frac{E_{2}}{E}\right)=\frac{P\left(E_{2}\right) P\left(\frac{E}{E_{2}}\right)}{P\left(E_{1}\right) P\left(\frac{E}{E_{1}}\right)+P\left(E_{2}\right) P\left(\frac{E}{E_{2}}\right)}
$$
$$
=\frac{p(1)}{(1-p) \frac{1}{5}+p(1)}=\frac{p}{\frac{1-p+5 p}{5}}=\frac{5 p}{1+4 p}
$$
$E_{2}=$ Student knows the answer
and $E=$ Student answer correctly
$\therefore \quad P\left(E_{1}\right)=1-p \Rightarrow P\left(E_{2}\right)=p$
$\Rightarrow \quad P\left(\frac{E}{E_{2}}\right)=1$ and $P\left(\frac{E}{E_{1}}\right)=\frac{1}{5}$
Note that, the probability that student did not know the answer randomly =The probability that student know the answer.
$$
\therefore \quad P\left(\frac{E_{2}}{E}\right)=\frac{P\left(E_{2}\right) P\left(\frac{E}{E_{2}}\right)}{P\left(E_{1}\right) P\left(\frac{E}{E_{1}}\right)+P\left(E_{2}\right) P\left(\frac{E}{E_{2}}\right)}
$$
$$
=\frac{p(1)}{(1-p) \frac{1}{5}+p(1)}=\frac{p}{\frac{1-p+5 p}{5}}=\frac{5 p}{1+4 p}
$$
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