i totally didn't believe this, so i wrote a small script to test it, and 'lo --pic.twitter.com/suTIkc9sH1
If you pick HHHT ex ante to compare with, the probability of getting it is the same as HHHH. It’s only if you pick the comparison ex post that the original claim is correct, for the same basic reason that a “garbage hand” in poker is more probable than a straight flush.
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no, it’s legit. Consider a random length-5 sequence. There are three ways to contain HHHH, but four ways to contain HHHT. So it’s more likely a length-5 string contains HHHT, even ex ante!
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We’re considering different cases. I was analyzing length-4 sequences, whereas you are considering subsequences of length 4 in length-5 sequences. The “three ways” and “four ways” you mention are the type of equivalence classes I was referring to here:https://twitter.com/mhartl/status/1222352805911875584 …
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Technically, any five-card hand in poker is equally probable, but the equivalence class of “garbage hands” is vastly larger than the equivalence class of straight flushes. Likewise, things that “look like” HHHT—like HTHH or HHTH—are more numerous than HHHH or TTTT.
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Our intuition is drawn to patterns with low information content (entropy): “jack-high straight flush in spades” completely specifies the hand, whereas a garbage hand’s cards have to be individually listed. The less information needed to specify a hand, the less probable it is.
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