Question

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

• A. $p=1,q=0$
• B. $p=2/3,q=1/2$
• C. $p=3/5,q=2/3$
• D. there are inifinitely many value of $p$ and $q$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $p=1,q=0$
B $p=2/3,q=1/2$
C $p=3/5,q=2/3$
D there are inifinitely many value of $p$ and $q$

Let $A,B$ and $C$ be the events that the student is successful in tests I, II and III, respectively. Then P (the student is successful)
$=P\left[\right(A\cap B\cap C^{\prime })\cup (A\cap B^{\prime }\cap C)\cup (A\cap B\cap C\left)\right]$
$=P(A\cap B\cap C^{\prime })+(A\cap B^{\prime }\cap C)+(A\cap B\cap C)$
$=P\left(A\right)P\left(B\right)P\left(c^{\prime }\right)+P\left(A\right)P\left(B^{\prime }\right)P\left(C\right)+P\left(A\right)P\left(B\right)P\left(C^{\prime }\right)$
$[\because$ $A,B$ and $C$ are independent]
$=pq(1-1/2)+p(1-q)\left(1/2\right)+\left(pq\right)\left(1/2\right)$
$=\frac{1}{2}[pq+p(1-q)+pq]=\frac{1}{2}p(1+q)$
$\therefore \frac{1}{2}=\frac{1}{2}p(1+q)\Rightarrow p(1+q)=1$
This equation is satisfied for all pairs of values in (a), (b) and (c). Also, it is satisfied for inifinitely many values of $p$ and $q$. For instance, when $p=n/(n+1)$ and $q=1/n$, where $n$ is any positive integer.