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## Monday, March 23, 2015

### Flippin' Probabilities

I suspect we all get a kick from mathematical problems that lead us merrily down wrong intuitive paths; statistics is often a source for such misdirecting problems. Mike Lawler pointed to a good one over the weekend. The problem arises in an old Peter Donnelly TEDTalk (below), starting at about the 4-minute mark (but his entire 22-minute talk is definitely worth a listen). The problem is simply this:

When flipping a fair coin repeatedly (heads/tails), which 3-part sequence is more likely to appear first:  HTH or HTT?
i.e., In a running sequence, is HTH likely to show up before HTT, or HTT before HTH, or, over many trials, are the probabilities equal?
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MANY people jump to the conclusion that the two possibilities are equally likely, no doubt thinking in terms of the true equal probability of each single flip being an 'H' or a 'T.'
BUT NO, the somewhat surprising answer is that the HTT sequence is more likely to occur first, and the explanation Donnelly gives (again starting at ~4-min. point), has to do with the 'clumping' of overlapping 3-part chunks:

It may be worth noting that one could identify 3-part sequences that DO have equal probability of occurrence, as well as additional sequences, which like the above two, have unequal probabilities.

Donnelly doesn't really expound too fully on the explanation for the outcome, so you may wish to follow along Mike Lawler's videos (linked to above), or read up on "Penney's Game" (named after creator Walter Penney) which relates to all this:
http://en.wikipedia.org/wiki/Penney%27s_game

The always-entertaining "Scam School" once did a 15-min. episode on Penney's Game here: