Question

The probability of guessing the correct answer to a question is $\left( {\dfrac{x}{2}} \right)$. If the probability of not guessing the correct answer to this question is $\dfrac{2}{3}$, then x equals:
(A) $2$
(B) $3$
(C) $\dfrac{2}{3}$
(D) $\dfrac{1}{3}$

Answer 1

Hint: The given question revolves around the concepts and principles of probability. One must know the basics of probability before attempting the given problem. Simplification rules and transposition are of great significance in solving such type of questions.

Complete step by step solution:
In the problem, we are given that the probability of not guessing the correct answer to a question is $\dfrac{2}{3}$.
Also, we are given that the probability of guessing the correct answer to a question is $\left( {\dfrac{x}{2}} \right)$.
Now, we know that the sum of the probabilities in a probability distribution table consisting of all possible ways of doing a certain thing is one. Hence, the sum of probability of happening and not happening of a certain thing is one.
Hence, the sum of probabilities of guessing the correct answer to a question and not guessing the correct answer to a question is one.
Let $P\left( {correct} \right)$ denote the probability of guessing the correct answer and $P\left( {not\,correct} \right)$ denote the probability of not guessing the correct answer.
So, we get, $P\left( {correct} \right) + P\left( {not\,correct} \right) = 1$
Now, we have $P\left( {correct} \right) = \left( {\dfrac{x}{2}} \right)$ and $P\left( {not\,correct} \right) = \dfrac{2}{3}$.
So, substituting the values of probabilities in the equation, we get,
$ \Rightarrow \dfrac{x}{2} + \dfrac{2}{3} = 1$
Now, we have to find the value of x from the above equation.
So, shifting all the constant terms to the right side of equation, we get,
$ \Rightarrow \dfrac{x}{2} = 1 - \dfrac{2}{3}$
Simplifying the expression further, we get,
$ \Rightarrow \dfrac{x}{2} = \dfrac{1}{3}$
Multiplying both the sides of the equation by $2$ , we get,
$ \Rightarrow x = \dfrac{2}{3}$
Hence, the value of x is $\dfrac{2}{3}$. So, option (C) is correct.

Note:
Probability is a measure of how likely an event is. The value is expressed from zero to one. The sum of all possible probabilities is always one. There are many ways of solving equation as the one formed in the question itself. Method of transposition involves doing the exact same thing on both sides of an equation with the aim of bringing like terms together and isolating the variable or the unknown term in order to simplify the equation and finding the value of the required parameter.