What is the probability that exactly three children in a family of five children will be boys? Assume that P(boy)=P(girl)
To calculate the probability of exactly three boys in a family of five children, we need to consider the number of ways we can have three boys out of five children.
First, let's determine all the possible combinations of three boys and two girls. We can use the binomial coefficient formula, also known as the combination formula, which is given by:
C(n, k) = n! / ((n-k)! * k!)
Where:
- n is the total number of children (5 in this case)
- k is the number of boys (3 in this case)
- n! represents the factorial of n
Using the formula, we can calculate the number of combinations:
C(5, 3) = 5! / ((5-3)! * 3!)
= 5! / (2! * 3!)
= (5 * 4 * 3!) / (2! * 3!)
= (5 * 4) / 2!
= 10
So, there are 10 possible combinations of three boys and two girls in a family of five children.
Since the probability of having a boy or a girl is assumed to be the same (0.5), we can raise this probability to the power of the number of boys in each combination (3 boys) and the power of the number of girls in each combination (2 girls) to get the probability of each combination:
P(boy)^3 * P(girl)^2 = (0.5)^3 * (0.5)^2 = 0.5^5 = 0.03125
Therefore, the probability that exactly three children in a family of five children will be boys is 0.03125, or 3.125%.