Suppose that we toss balls into four bins until some bin con

Suppose that we toss balls into four bins until some bin contains three balls. Each toss is independent, and each ball is equally likely to end up in any bin. What is the expected number of ball tosses?

Solution

as the balls are equally likely to end up in any ball and each ball in independent of the other then,

Let your number be F(0,0,0,0). There are fifteen numbers to work out:

F(0,0,0,0)=1+F(1,0,0,0)

F(1,0,0,0)=1+3/4F(1,1,0,0)+1/4F(2,0,0,0)

F(2,0,0,0)=1+3/4F(2,1,0,0)

and so on. The final one is F(2,2,2,2)=1F(2,2,2,2)=1.