A wanderer lives in a remote landscape with three villages: Avalon, Belthar and Cresthaven. Every morning, they leave the village they are in and travel to one of the other two villages, choosing between the two destinations at random with an equal 50–50 chance. If the wanderer begins in their home village of Avalon and stops after 100 days of travel, are they more likely to end up in Avalon or in Belthar? Or are both destinations equally likely?
The wanderer is slightly more likely to end up in Avalon. The longer they walk, the closer the probability of being in each village gets to an equal 1⁄3 apiece. But after even-numbered days, the wanderer will always have a marginally higher chance of ending in Avalon. And after odd-numbered days, they will have a higher chance of ending in Belthar (or Cresthaven).
In the problem setup, there is no difference between Belthar and Cresthaven—any argument we make for one can be made for the other. So the wanderer will always have the same probability of ending at each of those villages. This means that on any day where the wanderer has a greater than 1⁄3 probability of ending in Avalon, they necessarily have a less than 1⁄3 probability of ending at Belthar, and vice versa.
Furthermore, these two scenarios oscillate back and forth: any day in which the wanderer has a higher chance of ending at Avalon must be followed by a day in which they have a higher chance of ending at Belthar, and vice versa—if they’re more likely to be in a given village one day, they must be more likely to leave it the next day. This is most obvious on day one, where they begin at Avalon with 100 percent certainty and have a 50 percent chance of ending at Belthar (and a 0 percent chance of ending at Avalon). Now, because Belthar and Cresthaven link back to Avalon and the wanderer always changes villages, they are more likely to end up at Avalon than either of the other villages after day two, which ends in Avalon with a 50 percent chance and Belthar and Cresthaven with a 25 percent chance each. This is reversed the next day: because the wanderer is a little more likely to start the day at Avalon, they’re a little more likely to end it at one of the other villages. The difference in probability between the three villages goes down over time, and if the wanderer were to make infinite trips, the probabilities would converge to 1⁄3. But after any finite number, such as 100, the probabilities are still slightly lopsided toward or away from where the wanderer started.
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