Question

At the college entrance examination each candidate is admitted or rejected according to whether he has passed or failed the tests. Of the candidate 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

• A. $68$%
• B. $70$%
• C. $73$%
• D. $75$%

Answer 1

The correct option is A $68$%

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
$P\left(\frac{A}{B}\right)=\frac{P\left(A\right)P\left(\frac{B}{A}\right)}{P\left(A\right)P\left(\frac{B}{A}\right)+P\left(A^{\prime }\right)P\left(\frac{B}{A^{\prime }}\right)}=\frac{0.32}{0.32+0.15}=\frac{32}{47}=68$%