A student appears for tests I, II and III. The student is successful if the passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II and III are p,q and `(1)/(2)`, respectively. If the probability that the student is successful, is `(1)/(2)`, then

Options

A. `p=q=1`
B. `p=q=(1)/(2)`
C. `p=1,q=0`
D. `p=1,q=(1)/(2)`

Answer

C

Explanation

Let A , B and C denote the events of passing the tests I, II and III , respectively .
According to given condition ,
`=(1)/(2) P[(A cap B ) cup (A cap C)]`
`=P(A cap B )+P(A cap C ) -P(A cap B cap C) `
`=P(A) P(B) +P(A)*P(C) -P(A)*P(B) *P(C) `
`=pq+p*(1)/(2) -pq*(1)/(2) `
` implies 1 =2 pq +p-pq implies 1=p(q+1) `.......(i)
The values of option (c ) satisfy Eq . (i) .
[ Infact , Eq. (i) is satisfied for infinite number of values of p and q . If we take any value of q such that ` 0 le q le 1` , then p takes the value ` (1)/(q+1)` . It is evient that , ` o lt (1)/(q+1) le 1 ` i.e. ` 0 lt p le 1 `. But we have to choose correct answer from given ones . ]