Grinstead-Snell-Probability · Problem 11.1.5
Markov Chains
Mathematics · markov-chains-introduction
Problem
In Example 11.7, find the probability that the grandson of a man from Harvard went to Harvard.Intuitive explanation
This is a Markov chain problem where you're tracking how a property (college attendance) evolves across generations. The key insight is that each generation's college choice depends only on the previous generation's choice, not on the entire family history—so you multiply transition probabilities across two steps. To find the probability that a grandson attended Harvard given his grandfather did, you need to consider all possible paths the father could have taken, then weight each path by its likelihood of leading to Harvard in the next generation.
Intuitive explanation
This is a Markov chain problem where you're tracking how a property (college attendance) evolves across generations. The key insight is that each generation's college choice depends only on the previous generation's choice, not on the entire family history—so you multiply transition probabilities across two steps. To find the probability that a grandson attended Harvard given his grandfather did, you need to consider all possible paths the father could have taken, then weight each path by its likelihood of leading to Harvard in the next generation.