1. Hot Hand: Better than Chance
Analyzing Numbers in the News
Schield ICOTS
15 May 2008
2014
Hot Hand: Better than Chance
Milo Schield
Augsburg College
Editor:
VP, National Numeracy Network
US Rep: International Statistical Literacy Project
Elected Member: International Statistical Institute
2008SchieldNNN6up.pdf
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2. Miller & Sanjurjo: Thesis
Analyzing Numbers in the News
Schield ICOTS
15 May 2008
2014 2
Miller & Sanjurjo: Thesis
"We find a subtle but substantial bias in a standard measure of the conditional dependence of present outcomes on streaks of past outcomes in sequential data.
The mechanism is driven by a form of selection bias, which leads to an underestimate of the true conditional probability of a given outcome when conditioning on prior outcomes of the same kind.
2008SchieldNNN6up.pdf
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3. You are may be ignorant, but you are not stupid
Schield ICOTS
Analyzing Numbers in the News
15 May 2008
2014 3
You are may be ignorant, but you are not stupid
Chance of 2nd head following 1st head:
Greater than 50%
Equals 50%
Less than 50%
I have no idea
Which of these answers do you expect the paper to prove?
A >50% B =50% C < 50% D No idea
Which of these answers does this paper claim?
Answer: C (less than)
2014-Schield-ICOTS-6up
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4. Andrew Gelman “There really is a hot hand”
Analyzing Numbers in the News
Schield ICOTS
15 May 2008
2014 4
Andrew Gelman “There really is a hot hand”
“yup, I’m serious. Josh Miller came into my office yesterday and convinced me that the hot hand is real.”
Miller: “If he were to sample 1 million fair coins and flip each coin 4 times, observing the conditional relative frequency for each coin, on average the relative frequency [of heads following heads] would be approximately 0.4.”
2008SchieldNNN6up.pdf
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5. Four flips Chance of 2nd Head given 1st head with room for 2nd.
Schield ICOTS
2014
Four flips
Chance of 2nd Head
given
1st head with room for 2nd.
0.50 vs 0.40.
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6. Chance of Run | Start of Run Three flips: 0.500 vs 0.417
Schield ICOTS
2014
Chance of Run | Start of Run Three flips: vs 0.417 .
2014-Schield-ICOTS-6up

7. Andrew Gelman Why average by coin/player?
Analyzing Numbers in the News
Schield ICOTS
15 May 2008
2014 7
Andrew Gelman Why average by coin/player?
Josh M: “Does this mean that the, e.g., probability of having a girl be born after a boy isn’t exactly 50/50 either? Does it mean that having only children of the same sex is less likely than we thought?”
Andrew Gelman: “Josh: Pr(girl) = .485 (approximately), no matter what was the sex of the previous baby. But if you were to estimate the conditional probabilities in a certain way (averaging within families, then averaging across families, rather than weighting each birth equally in the average), then you could come up with a biased estimate. And this sort of thing can be tricky; recall the error-filled work of Satoshi Kanazawa.”
2008SchieldNNN6up.pdf
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8. Increase (Put two worst in one class): 8.5
Schield ICOTS
2014
Two Kinds of Averages
Three students. Scores of 6, 8 and 10.
Average student score by student: 8
If we group 2 students in one class, the 3rd in a second class, average student scores by class can:
Increase (Put two worst in one class): 8.5
Decrease (Put two best in one class): 7.5
Be same (Put top & bottom in one class): 8
2014-Schield-ICOTS-6up

9. Average chance by possible run is 0.500
Schield ICOTS
2014
Two Kinds of Averages
In computing chance of 2nd head given 1st head.
Average chance by possible run is 0.500
Average chance by coin is 0.40
Why average the per coin chances?
2014-Schield-ICOTS-6up

10. Andrew Gelman Why average by coin/player?
Analyzing Numbers in the News
Schield ICOTS
15 May 2008
2014 10
Andrew Gelman Why average by coin/player?
Alex: “Is there a simple / intuitive reason why one would average this way [by coin or by person], rather than say, vectorizing the matrix in the above example first and averaging once?”
Andrew: “ If you’re estimating a common probability such as Pr(girl), you’d average everybody. For the hot hand, the usual approach is to get a separate estimate for each player, hence the separate averaging.”
2008SchieldNNN6up.pdf
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11. Miller & Sanjurjo: Part II
Analyzing Numbers in the News
Schield ICOTS
15 May 2008
2014 11
Miller & Sanjurjo: Part II
[F]or those trials that immediately follow a streak of successes, observing that the relative frequency of successes is equal to the base rate of success, is in fact evidence in favor of the hot hand, rather than evidence against it.
ews_105#Does_selection_bias_explain_the_.22hot_hand.22. 3F
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12. It all depends on what you take into account!!! Conclusion 12
Analyzing Numbers in the News
Schield ICOTS
15 May 2008
2014 12
Conclusion
It all depends on
what you
take into account!!!
2008SchieldNNN6up.pdf
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13. Andrew Gelman “There really is a hot hand”
Analyzing Numbers in the News
Schield ICOTS
15 May 2008
2014 13
Andrew Gelman “There really is a hot hand”
After simulating this in R, Gelman said, “Hey! That's not 50%! Indeed, if you get this sort of data, it will look like people are anti-streaky (heads more likely to be followed by tails, and vice-versa), even though they're not.”
The classic 1985 paper by Gilovich, Vallone, and Tversky and various followups used these frequency comparisons, and as a result they all systematically underestimated streakiness, reporting no hot hand when, actually, when the data are analyzed correctly, the evidence is there…”
2008SchieldNNN6up.pdf
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14. Chance of 2nd Head Given 1st head & Room for 2nd
Schield ICOTS
2014
Chance of 2nd Head Given 1st head & Room for 2nd .
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15. Schield ICOTS
2014
Two Flips of a Fair Coin .
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16. Miller & Sanjurjo: Part I
Analyzing Numbers in the News
Schield ICOTS
15 May 2008
2014 16
Miller & Sanjurjo: Part I
Miller and Sanjurjo point out that there is selection bias inherent in observing a finite sequence after it has been generated: the first flip in a streak of heads will not figure in the proportion of heads that follow a head.
Since the overall proportion of heads in the sequence is 1/2, the proportion of heads that follow a head is necessarily less than 1/2. The third column computes for each sequence the relative frequency of a head following a head. This is what the paper calls a "conditional relative frequency",
2008SchieldNNN6up.pdf
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