Question

Assuming that for a husband wife couple the chances of their child being a boy or a girl are the same. Then what is the probability of their two children being a boy and a girl?
A.\[\dfrac{1}{4}\]
B.1
C.\[\dfrac{1}{2}\]
D.\[\dfrac{1}{8}\]

Answer 1

Hint: We have to find the probability that a couple has two children and one is a boy and another is a girl. The total outcomes, in this case, are 2. We will apply the binomial distribution formula to compute the required probability.

Formula used : We use the binomial distribution formula which is given by \[P(x) = {}^n{C_x} \cdot {p^x} \cdot {(1 - p)^{n - x}}\],
where \[n{\rm{ }} = \]Total number of events,
\[x{\rm{ }} = \]Total number of successful events
\[1{\rm{ }}-{\rm{ }}p{\rm{ }} = \]Probability of failure
\[p = \]Probability of success on a single trial.
The formula of combination is \[{}^n{C_r} = \frac{{n!}}{{r!\left( {n - r} \right)!}}\]

Complete Step-by-Step Solution:
Given that the chances that husband and wife have a child being a boy or a girl are the same,
From the given data,
Number of outcomes i,e., number of children required are 1 boy and 1 girl = 2 ,
Probability that the child is a girl = Probability that the child is a boy = \[\dfrac{1}{2}\],
Now we will use binomial distribution formula i.e.,\[P(x) = {}^n{C_x} \cdot {p^x} \cdot {(1 - p)^{n - x}}\],
Now we will substitute the values\[n = 2\],\[x = 1\],\[p = \dfrac{1}{2}\], in the formula
Probability of their two children being a boy and a girl \[ = {}^2{C_1} \cdot {\left( {\dfrac{1}{2}} \right)^1} \cdot {\left( {1 - \dfrac{1}{2}} \right)^{2 - 1}}\]
Now we will \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] the formula
\[ = \dfrac{{2!}}{{1!\left( {2 - 1} \right)!}}\left( {\dfrac{1}{2}} \right){\left( {\dfrac{1}{2}} \right)^1}\]
\[ = 2\left( {\dfrac{1}{2}} \right)\left( {\dfrac{1}{2}} \right)\] [Since \[2! = 2 \times 1 = 2\] and \[1! = 1\]]
\[ = 2\left( {\dfrac{1}{4}} \right)\]
\[ = \dfrac{1}{2}\]
Probability of their two children being a boy and a girl\[ = \dfrac{1}{2}\].

Hence the correct option is C.

Note:Binomial distribution summarizes the number of trials, or observations when each trial has the same probability of attaining one particular value. The binomial distribution determines the probability when the success and failure of an event are involved.