### A Simple Model of Real But Random-Looking Hot Streaks

This model is not supposed to be realistic!

Suppose:

Before every shot, a player enters either the state "hot streak" or "cold streak" with probability 1/2.

A player on a hot streak has a 2/3 probability of hitting a shot during that streak.

A player on a cold streak has a 1/3 probability of hitting a shot during that streak.

We can program this, and know with certainty that there are periods when the player has a 2/3 chance of making a shot, and periods when he has a 1/3 chance... and yet it does not help us at all in predicting the next shot. (From the outside, not knowing if the streak is "on" or not, there is always a 50% probability that the next shot will go in.)

Here is a Python program implementing this algorithm and also implementing another loop with a simple 50% chance of hitting for comparison:

import random

SHOTS = 50

print("Shooting with hot streaks:")
for shot in range(1, SHOTS):
hot = (random.random() < .5)
if hot:
make = (random.random() < .66)
else:
make = (random.random() < .33)
mark = 'X' if make else 'O'
print(mark, end='')
print("")

print("Shooting without hot streaks:")
for shot in range(1, SHOTS):
make = (random.random() < .5)
mark = 'X' if make else 'O'
print(mark, end='')
print("")

And here are some runs of the program:

1. I assume you are presenting this as a model of a player who would not be recognized as having a hot hand in the models that "de-bunked" the theory but nevertheless has a hot -hand. Is that correct ?

I think those model used a definition of hot hands where recent performance would be a predictive guide to outcome of the next shot.
You are clearly using a different definition - but no problem with that.

Your player would on average have just as many runs of successive baskets as a player who was never in or out the zone but just always scores 50% of the time - so I think you would have trouble persuading a basketball fan that this player was streaky and the 50% guy not - unless you showed them the source code.

If you tweaked the parameters of your program so that the player changed zones less frequently but still scored 50% of baskets then you would have a player that fans would start to consider streaky and would be recognized as such by the hot-hands analysis. I think this is the scenario that the original studies were trying to identify through statistical analysis and is (I think) how most people interested in the topic would define "sreakiness".

Finally: I do not think your views are in line with the author of "Willful Ignorance". Reading beyond the passage you quote I think he talks about how the possibility of understanding the apparent randomness of "hot hands" would allow opportunities of profiting from the predictive power this would give.

2. IN addition: In you model you have chosen parameters to demonstrate that it is possible to have "hot-hands" but for it to statistically undetectable from randomness.

Assuming hot-hands exists in the real world and that there are a myriad of parameters affecting it - isn't there going to be a vanishingly small chance that all the parameters line up so as to make the hot-hands effects look random ?

1. No, that is *exactly* what makes things look random: that there are a myriad of factors affecting the outcome! A child's height, .e.g., is not actually random, but is distributed around the parents' heights as though it were precisely because a myriad of factors affect it! That is just standard textbook stuff, not a novelty from me!

2. I agree with what you write here, but my point is rather that your model requires:

- Each individual event to be uninfluenced by the previous event (otherwise it will not seem appear random)
- Each individual event to have "states" that influence outcome such that people with knowledge of these states could in theory profit.

It just seems unlikely to me (though of course not impossible) that any real world event (like a basketball free throw) would have such "states (confidence level of player, adrenaline level, or whatever) that would never persist beyond a single event.

3. Rob, this is an existence proof. It does not have to be realistic. It's purpose is only to show that this is possible.

3. This does not address the question. In a hot streak the probability of success is affected by the prior result. In your simulation each event is independent.

1. No Ken: I am saying Tversky et al. *defined this in an arbitrary an misleading fashion*. So your demand that I make my model match their definition is a demand that I drop the central part of my complaint about their paper and actually use the definition that I think is arbitrary and misleading!

2. "In a hot streak the probability of success is affected by the prior result."

See, I am saying that they are WRONG in making this the definition of a hot streak. I have repeated that this is my problem with their paper again and again.

3. Gene, the event in your model is not 'taking a shot'v it is the print statement. And the chance of printing an X is 50% for each event and each event is independent. So there is no hot hand here. If you reposition the "are you hot" outside the loop, and follow a hot by a cold streak to get back to 50% then you have a model . But it will show the tell tale streakiness too.

4. Apparently you can't read the model. The chance of printing an X is sometimes 66% and sometimes 33%. You can easily see I am right if you were allowed to bet on the sequence at even odds against me.

And then you were allowed a choice: one of you will know the setting of the "hot" variable, and one of you won't.

Which would you pick?

The probability for sure is 50% from a position IGNORANT OF THE SETTING OF THE HOT VARIABLE. But if you were told that setting before the "shot" you would certainly NOT treat it as 50% and you would clean up against the person ignorant of the setting who saw it as 50%.

4. Doesn't the notion of a streak imply that it lasts for more than one shot?