Question

A couple has two children, the older of which is a boy. What is the probability that they have two boys?
A. $\dfrac{1}{2}$
B. $\dfrac{1}{3}$
C. $\dfrac{3}{4}$
D. None of these

Answer 1

Hint: For finding probability, use the formula, $P(A)=\dfrac{n(A)}{n(S)}$ where A is the event, P(A) is the probability of event A, n(A) is the number of outcomes which favours the event A and n(S) is the total number of outcomes that can occur. Here A is the condition that the second child is a boy.

Complete step-by-step answer:
In the question it is given that a couple has two children, the older of which is a boy and we have to find the probability that they have two boys. So, first let us see what probability is. Probability is the branch of mathematics concerning the numerical description of how likely an event is to occur. Probability is a number that lies between 0 and 1. If the probability is 0, then it means that the event is impossible whereas, if the probability is 1, it means that the event is certain to occur. The higher the probability, the more likely is it for the event to occur. A simple example of probability is the tossing of a fair (unbiased) coin. Since the coin is fair, the outcomes are both equally probable ‘heads’ or ‘tails’. The probability of ‘heads’ equals the probability of ‘tails’ and since no other outcomes are possible, the probability of either ‘heads’ or ‘tails’ is $\dfrac{1}{2}$ (which can be also written as 0.5 or 50%).
The formula for probability is $P(A)=\dfrac{n(A)}{n(S)}$ where A is the event, P(A) is the probability of event A, n(A) is the number of outcomes which favours the event A and n(S) is the total number of outcomes that can occur.
Now, in the question we know that the older one is a boy. So, the younger one has only two probabilities, that is, a girl or a boy.
So the total number of outcomes, n(S) = 2.
It is asked in the question to find the probability where the younger child is a boy which means, n(A) = 1.
So, the probability that both the children are boys is, $P(A)=\dfrac{n(A)}{n(S)}=\dfrac{1}{2}$.
Hence we get the probability of both the children to be boys as $\dfrac{1}{2}$.
So, option (A) is the correct answer.

Note: Students must take care not to make any mistakes in the formula of the probability. And also, must be careful while taking the number of favourable outcomes and the number of the total outcomes and must make proper substitution of these value in the formula for probability