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 lt p lt 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

Options

A. `(3p)/(4p+3)`
B. `(5p)/(3p+2)`
C. `(5p)/(4p+1)`
D. `(4p)/(3p+1)`

Answer

C

Explanation

Let `E_(1)` = Student does not know the answer
`E_(2)` = Student knows the answer
and E = Student answer correctly
`therefore P(E_(1))=1-p " " P(E_(2))=p`
`P((E )/(E_(2)))=1` and `P((E )/(E_(1)))=(1)/(5)`
Note that, the probability that student did not tick the answer randomly = The probability that student know the answer.
`therefore P((E_(2))/(E ))=(P(E_(2))P((E )/(E_(2))))/(P(E_(1))P((E )/(E_(1)))+P(E_(2))P((E )/(E_(2))))`
`=(p(1))/((1-p)(1)/(5)+p(1))=(p)/((1-p+5p)/(5))=(5p)/(1+4p)`