Thursday, October 28, 2010

Penney's Paradox

Want to win a few coins... The "Penney Paradox" is a very intriguing though less-discussed paradox than some of its more famous counterparts (it's named after its discoverer Walter Penney, though it is also often discussed using a penny as the working example).

If one flips a fair coin 3 separate times, there are 8 equally probable (heads/tails) triplet-results: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. In this game a first player selects one of these triplets, and then a second player chooses a different one. The coin is then flipped repeatedly until one of the selected triplets appears as a run and the player having chosen it wins the game (and coin). For example, if the chosen triplets are HTH and THT and the flips go THHHTH, the last three flips mean that HTH has won... the first triplet appearing matching a player's choice, wins.

One might first think that any one triplet is just as likely to occur as any other. However, upon reflection it will probably be clear that given a series of 4-or-more flips there are more ways for a triplet like say HTH to occur than the triplets TTT or HHH to appear. But what is far more intriguing is that NO MATTER what triplet the first player chooses, there are triplets that player #2 can select giving him/her a probabalistic edge of winning.
"Futility Closet" site mentioned this a few weeks back (and how player 2 can make his/her choice), but without elaborating much on how the mathematics of it works.

"" covers the math here:

Or you can check out a briefer treatment on Wikipedia here:

I should also mention that IF you do have Martin Gardner's "Colossal Book of Mathematics" on-hand he covers the subject well in his chapter 23 on "nontransitive paradoxes" (it is the "nontransitivity" of the relationships involved that result in the differential probabilities for the triplets).

1 comment:

Education Man said...

Great information. To be honest with you while I was in grade school I never had a teacher use ideas like this. Honestly if a teacher had used this game I probably would had learned probability differently. Thank you for the post and links to the wiki's they are very interested to read about.