Question

Assertion :At the college entrance examination each candidate is admitted or rejected according to whether he has passed or failed the tests. Of the candidates who are really capable, $80$% pass the test and of the incapable, $25$% pass the test. Given that $40$% of the candidates are really capable, then the proportion of capable college students is about $68$%. Reason: $P\left(\frac{A}{B}\right)=\frac{P\left(A\right)P\left(\frac{B}{A}\right)}{P\left(A\right)P\left(\frac{B}{A}\right)+P\left(A^{\prime }\right)P\left(\frac{B}{A^{\prime }}\right)}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $A$ be the event that a really able candidate passes the test and let $B$ be the event that any candidate passes this test. Then, we have,
$P\left(\frac{B}{A}\right)=0.8,P\left(\frac{B}{A^{\prime }}\right)=0.25,P\left(A\right)=0.4,P\left(A^{\prime }\right)=1-0.4=0.6$
By Baye's formula,
$P\left(\frac{A}{B}\right)=\frac{P\left(A\right)P\left(\frac{B}{A}\right)}{P\left(A\right)P\left(\frac{B}{A}\right)+P\left(A^{\prime }\right)P\left(\frac{B}{A^{\prime }}\right)}=\frac{0.32}{0.32+0.15}=\frac{32}{47}=68$%