Solution
Pass the Parcel
Show the problem again
Five friends A, B, C, D, E stand in a circle. A holds a ball. Every second, whoever holds the ball throws it to one of the other four friends, chosen uniformly at random. What is the expected number of throws until E first receives the ball? Give an exact number.
Worked solution
The answer is 4. Whoever holds the ball (as long as it is not E) has E among exactly four equally likely targets, so every throw reaches E with probability 1/4, independently of the past. The waiting time is geometric with success probability 1/4 and mean 4. On the complete graph the hitting time between any two distinct vertices is n − 1.
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.