Question

In a math’s 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. $p = q = \dfrac{1}{2}$
C. $p = 1,q = 0$
D. $p = 1,q = \dfrac{1}{2}$

Answer 1

We have to use the formulas of probability to solve this question. Section A is compulsory and any one of section B or section C is to be attempted. The probability that the student is successful is also given so we have to use these data and find the solution.

Complete step by step solution:
According to the question we have to find which of the following options is false, that is, find the value of p and q.
So, let the events A, B and C are the events of the sections in which students pass.
$P(x)$ denotes the probability of happening of event x
So, $P(A) = p$, $P(B) = q$ and $P(C) = \dfrac{1}{2}$
Now, the student will be successful if he is able to pass section A and either of section B or Section C.
Hence it can be represented in a probability as = (Probability of passing in A) multiplied by ((Probability of selecting either of section B or C) multiplied by (Probability of passing B +Probability of passing C))
$\Rightarrow P(A) \times (\dfrac{1}{2} \times (P(B) + P(C))$
$\Rightarrow p \times (\dfrac{1}{2}(q + \dfrac{1}{2})) = \dfrac{{pq}}{2} + \dfrac{p}{4}$
But Probability that the student is successful in the exam is equal to $\dfrac{1}{2}$
So the above equation is equal to $\dfrac{1}{2}$, so we can write it as
$\Rightarrow \dfrac{{pq}}{2} + \dfrac{p}{4} = \dfrac{1}{2}$
$\Rightarrow 2pq + p = 2$
Now, we have to check our options in this equation
Hence we find that only option four, that is, $p = 1,q = \dfrac{1}{2}$ is satisfying the equation.
Hence, option four is our answer.

Note:
At last we have to check our options as we have an equation of two variables but only one equation. To solve any equation of two variables we need two equations. We also understand the concept of probability to solve these types of questions.