The Hot Hand

Every basketball fan knows the scenario.  Some player has a “hot hand.”  He–or she–makes what seems like an extraordinary string of three-point shots.  The coaches and players change their offensive and defensive strategies, convinced that a player in the midst of a “hot streak” is more likely to make the next three-pointer than not.

The same thing happens in the stock market and in other gambling venues.  Sorry, you may not think the stock market is a gambling venue, so let’s suppose we’re in a casino.  A player at a slot machine will have a run of six or seven wins and think the machine is “hot.”  Or, they will have what seems like a long losing streak.  How likely is that?

If we reduce this to flipping a coin, we have some intuition about the likelihood of streaks.  The problem is that the popular intuition is wrong.  It even has a name: the gambler’s fallacy.

For example, if you flip a fair coin–one where heads and tails are equally likely–six times, what are the chances of a “streak” where all six flips are heads?  The  the chances of all six turning heads is easy to calculate: it’s  (1/2) to the sixth power, or (1/64) or 1.56%.   We’d all agree that’s an unlikely event.

If we flip the same coin seven times, then the sequence of six heads could start with either the first flip or the second flip.  This gives a two-step procedure for finding the chances of a streak of six heads in seven flips.

For the first step, suppose the streak starts with the first flip. Then the seven flips must one of the following two sequences:


Each of these happens with probability (1/2) to the seventh power or (1/128), so the chances of the sequence starting on the first flip is


For the second step, suppose the streak starts with the second flip, then we must have had a tail on the first flip, and the only way this can happen is  the sequence (THHHHHH).  This has  probability (1/2) to the seventh or (1/128).

These are the only two ways we can get a streak of six heads in seven tries, so the sum of the two probabilities gives the answer:

(1/64) + (1/128) = (3/128)=2.34%.

A similar analysis works for a streak of six heads in eight flips, except that it takes three steps instead of two.  As above, if the streak starts with the first flip, the chances are (1/64).

For the next two steps, think about where the first tail occurs.  If the first tail occurs on the first flip,  then we need to have a streak of six in the remaining seven flips.  But this is the “streak of six heads in seven flips” that we just calculated, so the probability of the streak occurring in the remaining seven flips is (3/128), and the probability for step 2 is

[Probability of 1st tail on flip 1]*[Probability of a streak of six in seven flips] =


 Finally, the first tail could appear on the second flip, meaning the streak starts on the third flip of the eight.  That would mean the first flip was heads, the second tails, followed by the streak, so this final case has probability


Adding all these together gives


for a string of six heads in eight flips.

Even better, this analysis gives a recursive way to calculate the odds of a streak of coin flips.  The formula and a computer program for doing the calculation are here (scroll down to the mathematician’s answer).

So, what happens if you flip the same coin 200 times? What are the chances of seeing at least one streak of either six consecutive heads or six consecutive tails?  That’s where the recursive formula and the computer program are helpful.  It turns out that the chances of a streak of length six in 200 coin flips is over 96%!  In fact, there is an 80% probability of getting a streak of length of seven in 200 trials, and a 54% chance of getting streak of length eight in 200 trials.  There are some tables here that graphically show these results for streaks in 200 and 1000 trials.

The moral here is that truly random data is streaky, with sequences of consecutive outcomes, and that these occur more frequently that your intuition suggests.

If you increase the number of trials to 1,000, then the odds of what seem, intuitively, to be extraordinary events can be surprising. For example, the chances are about two to one in favor of a streak of ten appearing in 1,000 coin flips.

Getting back to three-point shots, what does this imply about our intuition on the hot hand?  Well, there’s research on that.  Basically, using the ideas above to analyze data from the Philadelphia Sixers, the researchers show that the strings of three-point shots can be perfectly explained by random chance.  Here’s another source about strings of heads and tails, with revealing graphs and more detail.

I was thinking about this after my post yesterday–the one with the chart from the Noetics Institute.  There is controversy about the meaning of this data and the manner in which the researchers have analyzed it.  The graphs are striking because they play on our very human desire to find patterns.  However, just because a pattern seems intuitively unlikely doesn’t mean that it is unlikely.

That’s why I said I the “coincidence” theory seemed more plausible to me than a “global consciousness” theory.  Understand, I like the idea of a global consciousness. It might even be true.  But I’m not convinced that this data supports that concept.

The same notion of “hot hands” applies in other settings–the stock market, for example, where we see bubbles form and burst based on incorrect assessments of probability and risk.

Well, I’ve got to run.  I’m grading papers from my stats class, and they’re doing really well.  I don’t want to break their streak…oh, wait…


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