Probability & Statistics

Question

1. A family has n > 0 children with probability αp^n, where α≤p/(1-p). (a) What is the probability that the family has no children? (b) If each child is equally likely to be a boy or a girl (independently of each other), what proportion of families consists of k boys and any number of girls?

Answer 1

(a) P(0) = 1 - P(at least 1)

             

             = 1 - Σ_n=1^∞(apⁿ)

 

You can finish this by using a formula for a geometric series.

 

(b) P(k boys and n girls) = (1/2)^k(1/2)ⁿ

 

      Σ_n=0^∞P(k boys and n girls)

 

This will calculate the probability of exactly k boys.

 

Use this to get the proportion.

 

 

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