Question

A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II, III are p, q and $\frac { 1 } { 2 }$ respectively. If the probability that the student is successful is $\frac { 1 } { 2 }$, then

• A. p = 1, q = 0
• B. $p = \frac { 2 } { 3 } , q = \frac { 1 } { 2 }$
• C. There are infinitely many values of p and q
• D. All of the above

Answer 1

Answer: (C)

There are infinitely many values of p and q

Answer 2

Let A, B and C be the events that the student is successful in test I, II and III respectively, then P (the student is successful)

$= P ( A ) \cdot P ( B ) \cdot P \left( C ^ { \prime } \right) + P ( A ) \cdot P \left( B ^ { \prime } \right) \cdot P ( C ) + P ( A ) \cdot P ( B ) \cdot P ( C )$

[∵ A, B, C are independent]

$= p q \left( 1 - \frac { 1 } { 2 } \right) + p ( 1 - q ) \left( \frac { 1 } { 2 } \right) + p q \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 } p ( 1 + q )$

Ž $\frac { 1 } { 2 } = \frac { 1 } { 2 } p ( 1 + q )$ Ž $p ( 1 + q ) = 1$.

This equation has infinitely many values of p and q.