Nemo is playing hide-and-seek with his forgetful fish friend. There are three reefs that Nemo can hide in. Nemo decides to hide in Reef A. The game begins …
- If Nemo’s friend searches
- If she searches Reef B, she will take 4 minutes, not find him, and swim out.
- If she searches Reef C, she will take 5 minutes, not find him, and swim out.
Being a forgetful fish, if she doesn’t find Nemo, she will swim out and totally forget which reef she was just in! (Meaning she can go into the same reef over and over again). If Nemo’s friend randomly selects a reef to search at each turn (with equal chance), and continues until she finds him, what is the expected length of the game?
How long, on average, does it take to find Nemo?
Reveal Answer
Solution
There are many ways to solve this (some more complicated than others). My favorite, and easiest to understand, is to realize that, if Nemo’s friend does not find him, then she is exactly back to where she started. We applied this same technique when we looked into Gambler’s Ruin.
Let E represent the expected length of the game. At each ‘turn’ one of three things can happen, each with ⅓ probability.
If Nemo’s friend searches Reef A then it takes three minutes. If she searches Reef B then it wastes four minutes and then she can try again (4+E minutes). If she searches Reef C, then it’s five minutes plus the expected length of the game (5+E minutes).
The average length of the game is twelve minutes.