Question

Find the expected number of boys in a family with 8 children assuming the sex distribution to be equally probable.

Answer 1

Hint: Binomial distribution is a probability of a success and failure outcome in an experiment that is repeated multiple times. It has two possible outcomes. It has two variables $n$ and $p$. $n$ represents the number of times the experiment runs and $p$ represents the probability of the specific outcomes. In this problem use the result “Expected value of binomial distribution is $np$”.

Complete step-by-step answer:
In this problem we have to find the expected number of boys in a family with 8 children assuming the sex distribution be equally probable.
Here the sex distribution to be equally probable. So the distribution must satisfy binomial distribution.
We know that $n$ and $p$ are the parameters in Binomial distribution.
Let us consider $p=$ probability of having a boy in family and $q=$probability of not having a boy in family.
We have given that sex distribution is equal probable.
Therefore the probability of having a boy in family and probability of not having a boy in family are equal.
Hence $p=q$.
Also we know that the sum of probability of success and probability of failure is $1$.
So $p+q=1$.
Using the condition $p=q$ and $p+q=1$ we get $p+p=1$.
Adding the left hand side of the equation we get $2p=1$.
Multiplying both sides of the above equation by $\dfrac{1}{2}$ we get $p=\dfrac{1}{2}$.
As $p=q$ so $p=q=\dfrac{1}{2}$.
Total children in the family is $8$.
So $n=8$.
Also expected value of binomial distribution is $np$.
Using the above formula the expected number of boys $=8\times \dfrac{1}{2}=4$.
Hence the expected number of boys in a family with 8 children assuming the sex distribution to be equally probable is $4$.

Note: For such a question we first need to find which type of distribution exactly the problem. Also students must be aware of that distribution, its parameters and related formulas. We can also use logic to solve the question, we know that there can be either a boy or girl child in the family. It is given that total 8 children are there and the sex distribution is equally probable, from which we get that there can be 4 boys and 4 girls in the family.