Coin flips only 40% likely to flip heads after a heads.

From the Wall Street Journal
http://www.wsj.com/articles/the-hot-hand...
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The ‘Hot Hand’ Debate Gets Flipped on Its Head

The new paper, written by Joshua Miller and Adam Sanjurjo, begins with a riddle. Toss a coin four times. Write down what happened. Repeat that process one million times. What percentage of the flips after heads also came up heads?

The obvious answer is 50%. That answer is also wrong. The real answer is 40%—and the authors say their correction should alter years of thinking about the hot hand.

The fallacy of the hot hand was established in a classic 1985 study that has since become a part of the social-sciences canon. The paper’s conclusion—that the appearance of shooting streaks was a misreading of randomness—was so counterintuitive that many refused to believe it. The uproar hasn’t abated over the years, yet even the most promising follow-ups found only a tepid hand. The feeling that you can’t miss after making several shots in a row was still a “massive and widespread cognitive illusion,” as the Nobel Prize winner Daniel Kahneman has written.

Nobel laureates think about the hot hand because it’s a bias that shapes important decisions. For these academics, the hot hand isn’t an isolated basketball occurrence. It’s an accessible example of how human beings behave with consequences for almost every industry.

Now, though, comes the most intriguing argument that human intuition wasn’t wrong. A basketball player who shoots the same percentage after a streak of makes as he does after a streak of misses was long accepted as proof against the hot hand. Miller and Sanjurjo’s paper asserts it’s actually evidence of the opposite.

“People were right to believe the hot hand exists,” said Sanjurjo, an economist at the University of Alicante in Spain.

Their breakthrough is the surprising math of coin flips. Take the 14 equally likely sequences of heads and tails with at least one heads in the first three flips—HHHH, HTHH, HTTH, etc. Look at a sequence at random. Select any flip immediately after a heads, and you’ll see the bias: There is a 60% chance it will be tails in the average sequence.
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Notice this is not looking at conditional probabilities like,

P(2nd flip is heads GIVEN 1st flip is heads)

which must equal 50% by independence of the coin flips.

Rather, this is saying that if you flip a coin 4 times and IF there is at least one heads in the first 3 flips, THEN

P(randomly picked heads in the first 3 flips is followed by a heads) = 40%

Actually, it's 17/42 or about 40.48% if you do the calculation.

I suppose if statistics were taken in this way to disprove the "Hot Hand" theory then this observation could be important.

It matters how statistics are done.

PairTheBoard