Coin Flip Paradox

A gambler has a proposition for you.

Coin Flip

The gambler asks a third party to flip a coin three times where the two of you cannot see it, and write down the results. The gambler then asks the flipper, “Was either or both of the first and second flips heads?” The flipper says, “Yes.” The gambler asks the flipper to circle the result that came immediately after the heads or, if both the first and second flips were heads, to pick one of the heads at random, and circle the result after that heads. The gambler then says she will bet you even money that the result circled is tails.

You know that the odds of every fair coin flip are fifty-fifty regardless of what came before, so you reason that the odds are not against you if you take this bet. Are you correct?

Surprisingly, the odds are indeed against you. This was only recently discovered by Joshua Miller and Adam Sanjurjo (working paper). To see why, look at this chart of the possible outcomes when a coin is flipped three times:

Coin Flip Paradox


The first column shows the eight possible sequences of three coin flips, all of which are equally likely. The first two outcomes are eliminated because heads is not flipped first or second. The remaining six outcomes are then equally likely. At first, the chart does not seem to make sense. The middle columns show that heads follows heads the same number of times that tails follows heads, as we would expect, so why is the average percentage of times heads is followed by heads shown to be 41.7%?

We know that after eliminating the first two outcomes (the flipper would not have said “Yes” if the first two flips were tails), the other six outcomes are equally likely. That means that the probabilities must be averaged, and 250% ÷ 6 = 41.7%. In the HHH outcome, the HH sequence appears twice, but this only counts for 100% in 1 out of 6 outcomes, the same as the HH sequence appearing once in the THH outcome. This calls to mind other puzzles such as the Two Children Problem and the Monty Hall Problem in that the probability changes based on possibilities that have been eliminated.

The hot hand “fallacy”

Previous studies of the “hot hand fallacy” purported to show that there is no such thing as a “hot hand” in basketball or other sports, because analysis showed that players’ shooting percentage was no better after a streak of successful shots. The idea of a “hot hand” was thought to be a cognitive illusion, where fans found meaning in a string of successful shots that were really just random clusters. However, Miller and Sanjurjo suggest that there is a statistical “cold hand” that these studies did not account for. If a player’s shooting percentage is statistically likely to be lower after a streak, and they instead shoot at their normal level, they may have a hot hand after all.

Wait a minute…

Does this mean that if a coin is flipped and lands heads, then the next flip is likely to be tails? Is the gambler’s fallacy also not really a fallacy? No. Every flip of a fair coin is still fifty-fifty and is not influenced by any flips before it. Take another look at the chart above. Heads appeared after heads four times and tails appeared after heads four times. Betting on tails after a heads would yield exactly the results you would expect: a fifty-fifty chance.

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