Question

A coin, having probability p of landing heads, is continually flipped until at least one head and one tail have been flipped.  

Find the expected number of flips that land on tails.

Answer 1

let T is total number of tails ; E(T|first is head) =1 (since if first is head the process ends with first tails)

E(T|first is tail) =1+Expected number of tails till first head =1+(1/p-1) ; therefore

expected number of flips that land on tails E(T) =E(T|first is head)*P(1st head)+E(T|first is tail)*P(1st is tail)
E(T) =1*p+(1+(1/p-1))*(1-p)=