A coin sequence conundrum | Colin

Zeke and Monty play a game. They repeatedly toss a coin until either the sequence tail-tail-head (TTH) or the sequence tail-head-head (THH) appears. If TTH shows up first, Zeke wins; if THH shows up first, Monty wins. What is the probability that Zeke wins?

My first reaction to this question was, “It’s 50-50, right? It has to be 50-50.” But then a moment’s pause: “If it was 50-50, you wouldn’t be asking.”
Of course it isn’t 50-50

If you were to play the game, or simulate it, you’d find that TTH shows up first – on average – about two-thirds of the time. But that’s weird! Surely TTH and…

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