Question

A student appears for test I, II and III. The student is successful is he passes either in test I and II or test I and III. The probabilities of the student passing in test 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

The correct option is D

All of the above

Explanation for the correct answer:

Step 1: According to the question

Let $A,B,C$ be the event of student passing in the test I, II, III.

Here, $A,B,C$ are independent events.

The probabilities of students passing in the test I be $p$, test II be $q$ and test III be $\frac{1}{2}$.

Step 2: Finding the probability.

The probability that the student is successful = $P\left[\left(A\cap B\right)\cup (A\cap C)\right]$

$$\begin{gathered} \Rightarrow \frac{1}{2}=P\left[\left(A\cap B\right)\cup (A\cap C)\right] \\ \frac{1}{2}=P(A\cap B)+P(A\cap C)-P(A\cap B\cap C) \\ \frac{1}{2}=P\left(A\right)P\left(B\right)+P\left(A\right)P\left(C\right)-P\left(A\right)P\left(B\right)P\left(C\right) \\ \frac{1}{2}=pq+p\left(\frac{1}{2}\right)-pq\left(\frac{1}{2}\right) \\ 1=2pq+p-pq \\ 1=pq+p \\ 1=p(q+1)-----\left(1\right) \end{gathered}$$

Step 3: Substitute $p=1,q=0$

Now take, $p=1,q=0$ then

The equation $1$ becomes,

$$\begin{gathered} \Rightarrow 1(1+0)=1 \\ 1=1 \end{gathered}$$

Step 4: Substitute $p=\frac{2}{3},q=\frac{1}{2}$

Now take, $p=\frac{2}{3},q=\frac{1}{2}$then

The equation $1$ becomes,

$$\begin{gathered} \Rightarrow 1=\frac{2}{3}\left(\frac{1}{2}+1\right) \\ \Rightarrow 1=\frac{2}{3}\left(\frac{3}{2}\right) \\ \Rightarrow 1=1 \end{gathered}$$

Thus, the equation $1$ satisfied for the infinite number of values of $p$ and $q$.

Hence, option (D) All of the above is the correct answer