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Favorite intro prob game: You & I both pick length-3 sequences of H/T. Then we flip a coin and whoever’s seq comes first wins. All seqs are equally likely so we both have 50% chance to win, right? NOPE. If you pick first, I can pick seqs ensuring I’m 2x-7x more likely to win!
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The easiest case to think about is if you pick HHH. Then I pick THH & the only way for you to win is if we get HHH right off the bat: As soon as we get T, I’ve def won since the only way for you to win is to get 3 H, but by then I’ve already got my THH. https://en.m.wikipedia.org/wiki/Penney%27s_game …
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I love how this fellow's name was Penney
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@jben0 seen this? -
Thanks Dan! It is somewhat related to the hot hand selection bias via pattern overlap. We discuss the connection an the online supplement, the ECMA is now on OSF https://osf.io/sv9x2 , here is an excerpt:pic.twitter.com/rWTyEuqz8j
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The basic idea is that Prop(H| H)= # HH/ (# HH + # HT).
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Ps. Here is the full connection, from our Gambler's Fallacy paper: https://osf.io/m5xsk/ pic.twitter.com/TQyEB9nSVe
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Penney's game https://en.m.wikipedia.org/wiki/Penney%27s_game … but was it invented by Walter F Penney (who collaborated with Tom Lehrer) or his son Walter H?
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Interesting. Resembles the Monty Hall problem. Information affects the odds in surprising ways.
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But aren't most people fooled because you don't specify sliding matching, and they assume discrete sets of 3 flips? As soon as you explicitly say "starting in any position", the hint gives the game away.
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What if I choose the optimal sequence before you though?
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Took me a minute but once you realize the steps of the game are not independent (since it’s rolling), it makes sense.
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