Mathematics, statistic 150 (equalprobabilities)

Question

A family has  children. Assume that each child is as likely to be a boy as it is to be a girl. Find the probability that the family has  girls if it is known the family has at least one girl.

answer:

Answer 1

A key to this question is that the birth of each child is independent of the birth of any other child. So the probability of having a daughter or a son is 50% for each birth, regardless of what else has happened.

If the family already has a girl, then the probability of the fami,y having a girl is 100%, p = 1.

If the question is what is the probability that the family will have more than one girl, then that would be based on the probability that each new addition to the family might be a girl, or p = 0.5 for each additional child.

As the family grows with more children, the probability that there would be at least one more girl can be determined by expanding the polynomial 1 = (p + q)ⁿ, where n is the number of additional children the family has. In that situation, if p is the probability of having a girl, and q is the probability of having a boy, then the probability of having at least one more girl is 1 - qⁿ, where n is the number of additional children the family has.

This can also be demonstrated by using Pascal's triangle to show the probability of all combinations of boys and girls for any given number of children in the family.

Hope this helps.