Suppose families always have one, two, or three children, with probabilities 1/4, 1/2, and 1/4, respectively. Assume everyone eventually gets married and has children, then find the probability of a couple having exactly four grandchildren.

Explanation

Let the events be
A: exactly one child
B : exactly two children
C : exactly three children
E : couple has exactly 4 grandchildren
`thereforeP(A)=1/4,P(B)=1/2,P(C)=1/4`
`thereforeP(E)=P(A).P(E//A)+P(B).P(E//B)+P(C).P(E//C)`
Now when couple having oen child say D, then D must have four children, which is not possible.
When couple has two children, either each child has two children or one has three children and other has one child. When coulpe has three children, any two of children have one child each aand remaining has two children.
`P(E)=1/4xx0+1/2[{:(underbrace(((1)/(2))^(2)),+underbrace(1/4xx1/4xx2)),((2","2),(1","3)or(3","1)):}]+1/4[{:(underbrace((1/4xx1/4xx1/2))),(" "1","" "1","" "2):}]`
`=1/8+1/16+3/128+27/128`