On Monday, September 28, 2015, the Wall Street Journal published the article "The 'Hot Hand' Debate Gets Flipped On Its Head."

As a former basketball player and ongoing sports junkie (who happens to have a graduate degree in Statistics), I couldn't help but be drawn in by this topic. As anyone who has played basketball will tell you, the idea of a "hot hand" has long been regarded as part of the game, yet numerous statistical studies could find no evidence that the next shot from a "hot" shooter was more likely to go in or that the next shot from a "cold" shooter was more likely to miss.

Amazingly, a new paper written by Joshua Miller and Adam Sanjurjo, reverses decades of thinking and proves that the "hot hand" does indeed exist. Perhaps most remarkably, they used the same exact data from the earlier studies to arrive at the exact opposite conclusion!

How is this possible you might ask? It turns out that the sampling method from the earlier studies was biased. For instance, one study looked at the probability of a 50% shooter (which is analogous to flipping a coin) making the next shot after making 3 shots in a row. So when the researchers looked at a sequence of makes and misses for a given shooter, they were only observing the shots that occurred after 3 makes in a row. They theorized that because they were working with a 50% shooter, the probability of a make after any number of makes in a row should still be 50%.

If a shooter had a streak of 3 makes in a row, then by definition the observation that the researchers were interested in was always a miss (the 4th shot). If a shooter had a streak of 4 makes in a row, then by definition there were two observations that the researchers were interested in - a make (the 4th shot) and a miss (the 5th shot). For a shooter with a streak of 5 makes, the researchers would have observed two makes (the 4th and 5th shots) and a miss (the 6th shot).

Streaks of 3 are much more common than streaks of 4, and streaks of 4 are much more common than streaks of 5, and so on. Therefore, the sample is biased in favor of observing misses, and what Miller and Sanjurjo proved in their recent paper is that the probability of a make after observing 3 makes should have been 40%.

The original researchers observed in their data that the probability of a make on the 4th shot was essentially 50% after 3 makes in a row, and they erroneously concluded that there was no such thing as a "hot hand" - the 50% shooter was just as likely to have a miss as a make after 3 makes in a row. This was based on their faulty premise that they should have observed a 50% success rate on the 4th shot after 3 makes in a row.

But with the new perspective from Miller and Sanjurjo that takes the biased sampling into account, one can see that observing a success rate of 50% on the 4th shot attempt is about 10% higher than they should have expected from the biased sample - so the original data actually proved that the "hot hand" effect increased the likelihood of a make by 10 percentage points relative to the expected success rate of 40%.

The Wall Street Journal article contains one mistake. It erroneously states that the probability of observing "heads" on a coin flip after observing 3 "heads" in a row is 40%. This is a misinterpration of the study. The probability of getting "heads" on any coin flip is still 50% on any coin flip no matter what happened on the flips leading up to that. But if we only looked at 4-toss sequences and only looked at the flips that occurred after 3 straight "heads," then we would expect to see "heads" on only 40% of those throws due to the sampling bias. I verified these results through Monte Carlo testing, using randomly generated numbers to simulate coin flips.

So the next time you are on a hot streak in the gym, you can take your next shot with even more confidence knowing that the math is on your side!

Every day, we deal with situations in the life insurance and annuity marketplace where conventional wisdom is just as misleading as the old shooting studies which have now been debunked. If you want to have an independent and objective expert look at your situation, then you too might end up being amazed at the truths that will be uncovered!