Hi there,

The Problem:
In a family with 5 children, assuming the probability of having a boy or a girl is equally likely, what is the probability of having exactly 4 boys?

Solution:
This situation can be modeled by the Binomial Distribution because:

1: The number of observations n is fixed :: 5 children in the family.
2: Each observation is independent :: each child's gender is independent of the others.
3: Each observation represents one of two outcomes :: child is a girl or a boy.
4: The probability of "success" p is the same for each outcome :: probability of having a boy is always the same.

The formula for the Binomial Distribution is 

P(X = k) = [nCk] * [p^k] * [q^(n-k)]

What do all these variables mean?

For having exactly 4 boys: 
X = the random variable representing the event "the child is a boy".
n = the number of trials (number of children) = 5
k = the number of successful trials (It's a boy.) = 4
n-k = the number of unsuccessful trials (wrong answers) = 5-4 = 1
p = the probability of success, "having a boy" = 1/2 = 0.5
q = the probability of failure, "having a girl" = 1/2 = 0.5

nCk = the combination "n choose k," how many ways to choose k items from a set of n items = 5C4
nCK = n!/(k!(n-k)!) = 5!/(1! 4!)
P(X = k) = the probability that with random guessing there will be exactly k boys.

Your problem asks for the probability that there will be exactly 4 boys, P(k=4)

P(k=4) = [5C4] * [(0.5)^4] * [(0.5)^1]= [5] = 0.15625

The probability of having 4 boys in a family of 4 children is 0.15625.

If you have questions about this, or you want to check your answer, feel free to email me.

Mrs. Figgy
math.in.the.vortex@gmail.com