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An examination is attempted by 5000 graduates, 2000 post-graduates and 1000 doctorate holders. The probabilities that a graduate, a post graduate and a doctorate holder will pass the examination are $\frac{2}{3}, \frac{3}{4}, \frac{4}{4}$ respectively. If one of the examine passed the examination, then the probability that he is a post-graduate is
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Verified Answer
The correct answer is:
$\frac{45}{169}$
Consider the event
$E_1=$ graduate holder
$E_2=$ post-graduate holders
$E_3=$ doctorate holders
$A=$ passed examination
$P\left(E_1\right)=\frac{5}{8}, P\left(E_2\right)=\frac{2}{8}, P\left(E_3\right)=\frac{1}{8}$
$$
P\left(A / E_1\right)=\frac{2}{3}, P\left(A / E_2\right)=\frac{3}{4}, P\left(A / E_3\right)=4 / 5
$$
$\therefore$ Required probability
$$
\begin{gathered}
=P\left(E_2 / A\right)=\frac{P\left(E_2\right) \times P\left(A / E_2\right)}{P\left(E_1\right) \times P\left(A / E_1\right)+P\left(E_2\right) \times P A / E_2} \\
+P\left(E_3\right) \times P\left(A / E_3\right)
\end{gathered}
$$
$E_1=$ graduate holder
$E_2=$ post-graduate holders
$E_3=$ doctorate holders
$A=$ passed examination
$P\left(E_1\right)=\frac{5}{8}, P\left(E_2\right)=\frac{2}{8}, P\left(E_3\right)=\frac{1}{8}$
$$
P\left(A / E_1\right)=\frac{2}{3}, P\left(A / E_2\right)=\frac{3}{4}, P\left(A / E_3\right)=4 / 5
$$
$\therefore$ Required probability
$$
\begin{gathered}
=P\left(E_2 / A\right)=\frac{P\left(E_2\right) \times P\left(A / E_2\right)}{P\left(E_1\right) \times P\left(A / E_1\right)+P\left(E_2\right) \times P A / E_2} \\
+P\left(E_3\right) \times P\left(A / E_3\right)
\end{gathered}
$$
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