Question

In a couple's family history, male children dominate the offspring. This couple seeks to break the tradition and decides to have children until they have a desired number of girls. Let us suppose that there is no factor that indicates that the number of men in your family has biological causes and that the distribution of women and men in your family has been a mere coincidence.

a)Find the probability that the family will have four children if they have decided to have children until the first girl appears.

b) What is the probability that the couple will have at most 4 children if they only expect to have a girl?

c) Find the probability that the family will have six children if they have decided to have three girls.

d) What is the probability that the couple will have at most six children if they want to have three girls?

e)How many children does this family expect to have? (calculate the children expected by each family for the situation in part a), part d) and then generalize for a number r of girls)

f)How many children (boys) does this family expect to have? (calculate the boys that each family expects for the situation in part a), part c) and then generalize for a number r of girls)

Answer 1

We would be looking at the first 4 parts here as:

a) Probability of having 4 children is computed here as:

= Probability that the first 3 are males * Probability that the fourth child is a female

= 0.5³*0.5

= 0.5⁴ = 0.0625

Therefore 0.0625 is the required probability here.

b) Probability that the family would have at most 4 children

= 1 - Probability that the first 4 children are males

= 1 - 0.5⁴

= 0.9375

Therefore 0.9375 is the required probability here.

c) Probability that the family has six children if they have decided to have three girls is computed here as:

= Probability of having 2 girls in the first 5 births * Probability that the 6th child is a girl

Therefore 0.15625 is the required probability here.

d) Now the probability that the couple will have at most six children if they want to have three girls is computed here as:
= 1 - Probability of 0, 1 or 2 girls in the first 6 children

Therefore 0.65625 is the required probability here.