Question

In a maths paper there are 3 sections A, B & C. Section A is compulsory. Out of sections B & C a student has to attempt any one. Passing in the paper means passing in A & passing in B or C. The probability of the student passing in A, B & C are $p,q$ & $1/2$ respectively. If the probability that the student is successful is $1/2$ then, which of the following is false

• A. $p=q=1$
• B. $p=q=1/2$
• C. $p=1,q=0$
• D. $p=1,q=1/2$

Answer 1

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

The correct options are
A $p=q=1$
B $p=q=1/2$
C $p=1,q=0$

Let the events $A,B,C$ are the sections in which students pass.

$\Rightarrow P\left(A\right)=p,P\left(B\right)=q,P\left(C\right)=\frac{1}{2}$

The student will be successful if he able to pass the section $A$ and either of section $B$ or Section $C=$(Probability of passing in $A$)\times (Probability of selecting either of section $B$ or $C$ \times (Probability of passing $B$ +Probability of passing $C$))=$P\left(A\right)\times (\frac{1}{2}\times (P\left(B\right)+P\left(C\right))=p\times (\frac{1}{2}\times (q+\frac{1}{2}))=\frac{pq}{2}+\frac{p}{4}$

But Probability that student is successful in the exam $=\frac{1}{2}$

$\Rightarrow \frac{1}{2}=\frac{pq}{2}+\frac{p}{4}$

$\Rightarrow 2pq+p=2$

From the given option only Option $4$ i.e.,$p=1,q=\frac{1}{2}$ satisfies the equation.

Hence, options $1,2,3$ are wrong.