There was an interesting post and discussion on the NBA subreddit of Reddit on the Hot Hand phenomenon and whether or not it is a fallacy.

A Numberphile video on the topic:

An article on the topic:

https://www.scientificamerican.com/article/do-the-golden-state-warriors-have-hot-hands/

In some parts of the Numberphile video, Professor Lisa Goldberg emphasizes that issues of the “Law of Small Numbers,” which is described in the Scientific American article as:

Early in their careers, Amos Tversky and Daniel Kahneman considered the human tendency to draw conclusions based on a few observations, which they called the ‘‘law of small numbers’’.

when looking at the hot hand phenomenon, comes from the fact that we don’t get to see what happens after an H at the end of a sequence. Let a sequence be a string of shots of some length. A shot is either a make H or a miss T. So a sequence of 3 shots might be:

$$ HTH $$

A make, a miss, and then a make. So looking at that, we see that after the first H, we missed, which is evidence against the hot hand. We don’t care what happens after a miss, the T. We can’t see what happens after the last shot, which is a make. This is what’s noted as causing the “Law of Small Numbers.”

A moment from the Numberphile video illustrating the probabilities of H after an H for each possible sequence of 3 shots, and the average of those probabilities:

And here, this “Law of Small Numbers” causes the average probability of H’s after an H to be 2.5/6. When the sequence is a finite length, the probability of an H after an H (or a T after a T) is biased below 0.5. As the sequence gets longer and tends toward infinity, the probability of an H after an H (or a T after a T) goes toward 0.5.

While all this is true, let’s look a little closer at what’s going on in this illustration to understand why and how exactly this bias occurs.

All possibilities of sequences of 3 shots:

$$ n = \textrm{3} $$

$$ \textrm{Average probability} = \frac{2.5}{6} = 0.416\bar{6} $$

Assuming that an H and a T each appear with 0.5 probability and there is no memory, i.e. no hot hand, each of the above 8 sequences are equally probable. The average probability of the 6 cases where we can evaluate where there is a hot hand or not (cases that have an H in the first or second shot) is calculated to be 2.5/6 < 0.5.
**But let’s count the number of H’s and T’s in the second column. There are 4 H’s and 4 T’s!** So we have:

$$ \frac {\textrm{Number of H’s}}{\textrm{Number of H’s & T’s}} = \frac {4}{8} = 0.5 $$

So it’s as if we’ve undercounted the cases where there are 2 shots that are “hot hand evaluations,” the last two sequences at the bottom of the list. In all (8) sequences of length 3, how many hot hand evaluations in total were there? (How many H’s or T’s in the 2nd column?) 8. How many of those were H’s? 4. So we have a hot hand make probability of 0.5.

It doesn’t necessarily mean that the way they counted hot hand makes in the Numberphile video is wrong. It’s just a particular way of counting it that causes a particular bias. It also may be the particular way the human instinct feels hot handedness – as an average of the probability of hot hand makes over different sequences. In other words, that way of counting may better model how we “feel” or evaluate hot handedness in real world situations.

**So why is the average probability over sequences < 0.5?**

When we evaluate hot-handedness, we are looking at shots that come after an H. Suppose we write down a list or table of each possible permutation of shot sequences of length \(n\) from less H’s, starting from the sequence of all T’s, down to more H’s, ending with the sequence of all H’s. We noted above that if we count all the hot hand makes H’s in all sequences (the H’s in the 2nd column), the probability of hot hand H’s among all hot hand evaluations (the number of H’s or T’s in the 2nd column) is 1/2. When we look at the list of sequences, what we notice is that a lot of the hot hand H’s (the 2nd column) are concentrated in the lower sequences toward the bottom. But these sequences heavy in so many H’s only give one probability entry in the 3rd column of 1 or near 1.

$$ n = \textrm{4} $$

$$ \textrm{Average probability} = \frac{5.6\bar{6}}{14} \approx 0.405 $$

$$ n = \textrm{5} $$

$$ \textrm{Average probability} = \frac{12.25}{30} \approx 0.408\bar{3} $$

Assuming equal probability of H and T on any given shot and no memory between shots: the entire list of sequences (the 1st column) will have an equal number of H’s and T’s. Additionally, all the hot hand evaluations (the 2nd column) will have an equal number of H’s and T’s.

Looking at the 1st column, we go from more T’s at the top to more H’s at the bottom in a smooth manner. Looking at the 2nd column though, we go from rows of T’s and as we go down we find that a lot of H’s are “bunched up” towards the bottom. But remember that we have a “limited” number of H’s in the 2nd column as well, namely 50% of all hot hand evaluations are H’s and 50% are T’s.

**Let’s look closely at how the pattern in the 1st column causes more H’s to be bunched up in the lower sequences in the 2nd column, and also if there is any pattern to the T’s when we look across different sequences.**

Higher sequences have less H’s (looking at the 1st column), which means more HT’s in those sequences as well, i.e. more hot hand misses. Lower sequences have more H’s, which means more HH’s in those sequences, i.e. more hot hand makes. This means that, looking at the 2nd column, higher sequences have more T’s and lower sequences have more H’s. **Lower sequences “use up” more of the “limited amount” of H’s** (limited because the number of H’s and T’s in the 2nd column are equal). Thus, H’s in the 2nd column are “bunched up” in the lower sequences as well. This causes there to be less sequences with higher probability (the 3rd column) than sequences with lower probability. Perhaps this is what brings the average probability below 0.5.

A naive look of the 2nd column shows that the highest sequences have a lone T as its hot hand evaluation, and many other hot hand evaluations of higher sequences end with a T. This makes sense since if a sequence consists of a lot of T’s, any H’s in it are unlikely to be the last two shot in the sequence, like …HH, which is what’s needed for the hot hand evaluations in the 2nd column to end with an H. And as long as a T is the last shot, the hot hand evaluation of the sequence will end with a T, since any lone H or streak of H’s in the sequence will have encountered a T as the next shot either with that last T shot in the sequence (…HHT) or meeting the first of consecutive T’s that lead up to the last T shot of the sequence (…HHTT…T).

Let’s divide up all the sequences in the 1st column into categories of how a sequence ends in its last 2 shots and use that to interpret what the last hot hand evaluation will be in the 2nd column for that sequence category. There are 4 possible ways to have the last 2 shots: TT, TH, HT, and HH. If a sequence ends in …TT, that “…” portion is either all T’s or if it has any H’s, we know that that sequence ends in a T before or at the second-to-last T in the sequence (either …H…TTT or …HTT). So in all cases but one (where the entire sequence is T’s and so there is no hot hand evaluation for the 2nd column), the last hot hand evaluation in the 2nd column will be a T. If a sequence ends in …TH, the thinking is similar to the case that ends in …TT since the very last H doesn’t provide us with an additional hot hand evaluation since the sequence ends right there, so the 2nd column also ends in a T. If a sequence ends in …HT, the last T there is our last hot hand evaluation, so the 2nd column also ends in a T. If a sequence ends in …HH, then the 2nd column ends in an H. So about 3/4 of all sequences end their 2nd column with a T. (\(3/4)n-2\) to be exact, since the sequences of all T’s and \((n-1)\) T’s followed by an H don’t have any hot hand evaluations.) **Thus, the T’s in the 2nd column are “spread out more evenly” across the different sequences** since (\(3/4)n-2\) of all sequences have a T for its last hot hand evaluation (the 2nd column), while the H’s are “bunched up” in the lower sequences. Thus, a relatively large number of sequences, especially sequences that are higher up, have their probabilities (the 3rd column) influenced by T’s in the 2nd column, bringing the average probability across sequences down.

$$ n = \textrm{6} $$

$$ \textrm{Average probability} \approx 0.4161 $$

**As \( n \) grows larger, the average probability seems to drift up. **

Looking at the top of the list of sequences for \( n = 4 \), there are 3 sequences with a 0 in the 3rd column. These 3 sequences consist of 1 H and 3 T’s (and TTTH is uncounted because there is no hot hand evaluation in that sequence). At the bottom, we have the HHHH sequence giving a 1 in the 3rd column, and then 4 sequences that have 3 H’s ant 1 T. The entries in the 3rd column for these 4 sequences are 1, 0.5, 0.5, and 0.667.

For sequences of \( n = 5 \), there are then 4 sequences at the top of the list that give a 0 in the 3rd column. At the bottom, the HHHHH sequence gives a 1 in the 3rd column, and then the sequences with 4 H’s and 1 T give 1, 0.667, 0.667, 0.667, 0.75 in the 3rd column.

For sequences of \( n = 6 \), there are then 5 sequences at the top of the list that give a 0 in the 3rd column. At the bottom, the HHHHHH sequence gives a 1 in the 3rd column, and then the sequences with 5 H’s and 1 T give 1, 0.75, 0.75, 0.75, 0.75, 0.8 in the 3rd column.

This pattern shows that as \( n \) increases, we get \( (n – 1) \) sequences at the top of the list that always give 0’s in the 3rd column. At the bottom there is always 1 sequence of all H’s that gives a 1 in the 3rd column. Then for the sequences with \( (n – 1) \) H’s and 1 T, we always have 1 sequence of THH…HH that gives a 1 in the 3rd column, then \( (n – 2) \) sequences that give a \( \frac{n – 3}{n – 2} \) in the 3rd column, and always 1 sequence of HH…HT that gives a \( \frac{n – 2}{n – 1} \) in the 3rd column. So as \( n \) becomes large, the entries in the 3rd column for these sequences with \( (n – 1) \) H’s and 1 T get closer to 1. For small \(n\), such as \(n = 3\), those entries are as low as 0.5 and 0.667. But the entries in the 3rd column for the sequences high in the list with 1 H and \( (n – 1) \) T’s remain at 0 for any \(n\). Thus, as \( n \) becomes large, the lower sequence entries in the 3rd column become larger, shifting the average probability over sequences up.

Roughly speaking, when we only have one shot make in a sequence of shots (only 1 H among \(n-1\) T’s), we have only one hot hand evaluation possible, which is the shot right after the make. Ignoring the case of TT…TH, that hot hand evaluation coming after the H will always be a miss. **Thus, when there is only one shot make in a sequence, the hot hand probability is always 0.** On the other hand, when we have only one shot miss in a sequence, ignoring the TH…HH case, we will have 1 hot hand miss and many hot hand makes. **Thus, our hot hand probability in these sequences with only 1 T will always be less than 1, and approaches 1 as \( n \) approaches \( \infty \).** In a rough way, this lack of balance between the high sequences and low sequences drags down the average probability over the sequences below 0.5, with the amount that’s dragged down mitigated by larger and larger \( n \).

A possible interesting observation or interpretation of this is **how it might lead to the human mind “feeling” the gambler’s fallacy (e.g. consecutive H’s means a T “has to come” soon) and the hot hand fallacy (e.g. consecutive H’s means more H’s to come)**. The above results show that in finite length sequences, when a human averages in their mind the probability of hot hand instances across sequences, i.e. across samples or experiences, the average probability is < 0.5. In other words, across experiences, the human mind "feels" the gambler's fallacy, that reversals after consecutive results are more likely. But when a human happens to find themselves in one of the lower sequences on a list where there are relatively more H's than T's in the 1st column, what happens is that the hot hand evaluations (the 2nd column) are likely to have a lot more H's than what you'd expect, because H's are "bunched up" towards the bottom of the 2nd column. What you expect are reversals - that's what "experience" and the gambler's fallacy that results from that experience tells us. But when we find ourselves in a sequence low in the list, the hot hand instances (the 2nd column) give us an inordinately high number of hot hand makes because H's are bunched up towards the bottom of the list. So when we're hot, it feels like we're really hot, giving us the hot hand fallacy.
An actually rigorous paper on this subject, also found in a comment from the Reddit post, is *Miller, Joshua B. and Sanjurjo, Adam, Surprised by the Gambler’s and Hot Hand Fallacies? A Truth in the Law of Small Numbers*. One of the proofs they present is a proof that the average probability of hot hand makes across sequences is less than the probability of a make (i.e. using our example, the average of the entries in the 3rd column is less than 0.5, the probability of an individual make).