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  • Island-hopping problem

    I'm working through Ghahramani's text and am stuck on a simple bridge-hopping problem (page 45, number 21 of the International Edition if anyone has the book). I've gotten the answer using a tree diagram, but it seems as if there must be a simpler (faster!) way that I am missing. I'd appreciate any input!

    Here is the problem:
    "A salesperson covers islands A, B, ..., I. These islands are connected by the bridges shown in Figure 2.3. While on an island, the salesperson takes one of the possible bridges at random and goes to another one. She does her business on this new island and then takes a bridge at random to go to the next one. She continues this until she reaches an island for the second time on the same day. She stays there overnight. If she starts her trip from Island I tomorrow, in what percent of all possible trips will she end up staying overnight again at Island I?"

    Here is the schematic - the vertical lines connect AH, BI, CD, HG, IF, DE I just can't get them to align properly:

    A -- B -- C
    | | |
    H -- I -- D
    | | |
    G -- F -- E
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