You and I go out to eat at a conveyer-belt sushi restaurant. It serves four types of sushi: ahi roll, Boston roll, caterpillar roll and dragon roll. We each have access to our own conveyer belt that the chef loads continuously at random. Every plate has one piece of sushi on it, and each costs the same amount. We agree that I will eat every piece of sushi on my belt until I get two ahi rolls in a row, while you will eat every piece of sushi on your belt until you get an ahi roll followed immediately by a Boston roll. Whose meal do you expect to cost more?
My meal (ending in “ahi, ahi”) is expected to cost more. It seems like our situations are probabilistically equivalent. Because the chef loads the plates at random, ahi is just as likely to appear on my belt as it is on yours. Furthermore, once ahi has appeared on my belt, I have a one-in-four chance of finishing on the next plate, and the same goes for you.
The difference between our scenarios is what happens immediately after an ahi plate. For me, one of two things happens:
The next plate is ahi, and I’m finished.
The next plate is Boston/caterpillar/dragon, and I start from scratch, waiting for another ahi to come around.
For you, one of three things happens:
The next plate is Boston, and you’re finished.
The next plate is caterpillar/dragon, and you start from scratch, waiting for another ahi to come around.
The next plate is ahi, and you get another immediate shot at finishing if the subsequent plate is Boston.
This little advantage speeds up the expected time for you to finish your meal. In fact, you’re expected to eat 16 pieces of sushi before finishing while I’m expected to eat 20.
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