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==Warren Buffett's billion dollar gamble==


In a [http://www.quickenloans.com/blog/quicken-loans-billion-dollar-bracket-challenge widely publicized announcement] on January 21, 2014, Quicken Loans is offering a billion (sic) dollar prize to any contestant who can fill out the bracket perfectly in the March NCAA basketball tournament. Because they don't have a spare billion in the bank, they have insured against the possibility of a winner with Berkshire Hathaway (BH), paying an undisclosed premium believed to be around 10 million dollars. Is this is good deal for BH?
Some relevant data: to win you must predict all 63 game winners correctly. The number of entries is limited to 10 million. The prize is actually 500 million cash (or 1 billion over 40 years). Presumably Warren Buffett asked his actuary "are you very confident that the chance of someone winning is considerably less than 1/50". How would you have answered?


I put this forward as an interesting topic for open-ended classroom discussion. First emphasize that the naive model (each entry has chance 1 in  2<sup>63</sup>  to win) is ridiculous. Then elicit the notions that a better model might involve some combination of
==Forsooth==


*modeling typical probabilities for individual games
==Quotations==
*modeling the strategies used by contestants
“We know that people tend to overestimate the frequency of well-publicized, spectacular
*empirical data from similar past forecasting tournaments.
events compared with more commonplace ones; this is a well-understood phenomenon in
the literature of risk assessment and leads to the truism that when statistics plays folklore,
folklore always wins in a rout.”
<div align=right>-- Donald Kennedy (former president of Stanford University), ''Academic Duty'', Harvard University Press, 1997, p.17</div>


Here are two of many possible lines of thought.
----
(1) The arithmetic
 
<center>
"Using scientific language and measurement doesn’t prevent a researcher from conducting flawed experiments and drawing wrong conclusions — especially when they confirm preconceptions."
(5 million) × (3/4)<sup>63</sup> ≈ 1/14
 
</center>
<div align=right>-- Blaise Agüera y Arcas, Margaret Mitchell and Alexander Todoorov, quoted in: The racist history behind facial recognition, ''New York Times'', 10 July 2019</div>
suggests that if half the contestants are able to consistently predict game winners with chance 3/4, then it's a bad deal for BH. Fortunately for BH this scenario seems inconsistent with past data. Because the same calculation, applied to entries in a similar (but only 1 million dollar prize) ESPN contest last year, says that about
 
<center>
==In progress==
(4 million) × (3/4)<sup>32</sup> ≈ 1/14
[https://www.nytimes.com/2018/11/07/magazine/placebo-effect-medicine.html What if the Placebo Effect Isn’t a Trick?]<br>
</center>
by Gary Greenberg, ''New York Times Magazine'', 7 November 2018
entries should have predicted all 32 first-round games correctly. But none did (5 people got 30 out of 32 correct).
 
[https://www.nytimes.com/2019/07/17/opinion/pretrial-ai.html The Problems With Risk Assessment Tools]<br>
by Chelsea Barabas, Karthik Dinakar and Colin Doyle, ''New York Times'', 17 July 2019
 
==Hurricane Maria deaths==
Laura Kapitula sent the following to the Isolated Statisticians e-mail list:
 
:[Why counting casualties after a hurricane is so hard]<br>
:by Jo Craven McGinty, Wall Street Journal, 7 September 2018
 
The article is subtitled: Indirect deaths—such as those caused by gaps in medication—can occur months after a storm, complicating tallies
Laura noted that
:[https://www.washingtonpost.com/news/fact-checker/wp/2018/06/02/did-4645-people-die-in-hurricane-maria-nope/?utm_term=.0a5e6e48bf11 Did 4,645 people die in Hurricane Maria? Nope.]<br>
:by Glenn Kessler, ''Washington Post'', 1 June 2018
 
The source of the 4645 figure is a [https://www.nejm.org/doi/full/10.1056/NEJMsa1803972 NEJM article].  Point estimate, the 95% confidence interval ran from 793 to 8498.
 
President Trump has asserted that the actual number is
[https://twitter.com/realDonaldTrump/status/1040217897703026689 6 to 18].
The ''Post'' article notes that Puerto Rican official had asked researchers at George Washington University to do an estimate of the death toll.  That work is not complete.
[https://prstudy.publichealth.gwu.edu/ George Washington University study]
 
:[https://fivethirtyeight.com/features/we-still-dont-know-how-many-people-died-because-of-katrina/?ex_cid=538twitter We sttill don’t know how many people died because of Katrina]<br>
:by Carl Bialik, FiveThirtyEight, 26 August 2015
 
----
[https://www.nytimes.com/2018/09/11/climate/hurricane-evacuation-path-forecasts.html These 3 Hurricane Misconceptions Can Be Dangerous. Scientists Want to Clear Them Up.]<br>
[https://journals.ametsoc.org/doi/abs/10.1175/BAMS-88-5-651 Misinterpretations of the “Cone of Uncertainty” in Florida during the 2004 Hurricane Season]<br>
[https://www.nhc.noaa.gov/aboutcone.shtml Definition of the NHC Track Forecast Cone]
----
[https://www.popsci.com/moderate-drinking-benefits-risks Remember when a glass of wine a day was good for you? Here's why that changed.]
''Popular Science'', 10 September 2018
----
[https://www.economist.com/united-states/2018/08/30/googling-the-news Googling the news]<br>
''Economist'', 1 September 2018
 
[https://www.cnbc.com/2018/09/17/google-tests-changes-to-its-search-algorithm-how-search-works.html We sat in on an internal Google meeting where they talked about changing the search algorithm — here's what we learned]
----
[http://www.wyso.org/post/stats-stories-reading-writing-and-risk-literacy Reading , Writing and Risk Literacy]
 
[http://www.riskliteracy.org/]
-----
[https://twitter.com/i/moments/1025000711539572737?cn=ZmxleGlibGVfcmVjc18y&refsrc=email Today is the deadliest day of the year for car wrecks in the U.S.]
 
==Some math doodles==
<math>P \left({A_1 \cup A_2}\right) = P\left({A_1}\right) + P\left({A_2}\right) -P \left({A_1 \cap A_2}\right)</math>
 
<math>P(E)   = {n \choose k} p^k (1-p)^{ n-k}</math>
 
<math>\hat{p}(H|H)</math>
 
<math>\hat{p}(H|HH)</math>
 
==Accidental insights==
 
My collective understanding of Power Laws would fit beneath the shallow end of the long tail. Curiosity, however, easily fills the fat end.  I long have been intrigued by the concept and the surprisingly common appearance of power laws in varied natural, social and organizational dynamics.  But, am I just seeing a statistical novelty or is there meaning and utility in Power Law relationships? Here’s a case in point.
 
While carrying a pair of 10 lb. hand weights one, by chance, slipped from my grasp and fell onto a piece of ceramic tile I had left on the carpeted floor. The fractured tile was inconsequential, meant for the trash.
<center>[[File:BrokenTile.jpg | 400px]]</center>
As I stared, slightly annoyed, at the mess, a favorite maxim of the Greek philosopher, Epictetus, came to mind: “On the occasion of every accident that befalls you, turn to yourself and ask what power you have to put it to use.”  Could this array of large and small polygons form a Power Law? With curiosity piqued, I collected all the fragments and measured the area of each piece.


(2) The optimal strategy, as intuition suggests, is to predict the winner of each game to be the team you think (from personal opinion or external authority) more likely to win. For various reasons, not every contestant does this. For instance, as an aspect of a general phenomenon psychologists call [http://en.wikipedia.org/wiki/Probability_matching probability matching], a contestant might think that because some proportion of games are won by the underdog, they should bet on the underdog that proportion of times. And there are other reasons (supporting a particular team; personal opinions about the abilities of a subset of teams) why a contestant might predict the higher ranked team in most, but not all, games. So let us imagine, as a purely hypothetical scenario, that each contestant predicts the higher-ranked team to win in all except ''k'' randomly-picked games. Then the chance that someone wins the prize is about
<center>
<center>
Pr(in fact exactly ''k'' games won by underdog) × [(10 million) / <math>\tbinom{63}{k} </math> ]
{| class="wikitable"
 
|-
! Piece !! Sq. Inches !! % of Total
|-
| 1 || 43.25 || 31.9%
|-
| 2 || 35.25 ||26.0%
|-
|  3 || 23.25 || 17.2%
|-
| 4 || 14.10 || 10.4%
|-
| 5 || 7.10 || 5.2%
|-
| 6 || 4.70 || 3.5%
|-
| 7 || 3.60 || 2.7%
|-
| 8 || 3.03 || 2.2%
|-
| 9 || 0.66 || 0.5%
|-
| 10 || 0.61 || 0.5%
|}
</center>
</center>
provided the bracketed term is ≪1. This term is ≈ 0.15 for ''k'' = 6 and ≈ 0.02 for ''k'' = 7. The first term cannot be guessed -- as a student project one could get data from past tournaments to estimate it -- but is surely quite small for ''k'' = 6 or 7. This suggests a worst-case hypothetical scenario from BH's viewpoint: that an unusually small number of games are won by the underdog, and that a large proportion of contestants forecast that most games are won by the higher-ranked team. But even in this worst case it seems difficult to imagine the chance of a winner becoming anywhere close to 1/50.
<center>[[File:Montante_plot1.png | 500px]]</center>
The data and plot look like a Power Law distribution. The first plot is an exponential fit of percent total area. The second plot is same data on a log normal format. Clue: Ok, data fits a straight line.  I found myself again in the shallow end of the knowledge curve. Does the data reflect a Power Law or something else, and if it does what does it reflect?  What insights can I gain from this accident? Favorite maxims of Epictetus and Pasteur echoed in my head:
“On the occasion of every accident that befalls you, remember to turn to yourself and inquire what power you have to turn it to use” and “Chance favors only the prepared mind.”


'''Other estimates'''
<center>[[File:Montante_plot2.png | 500px]]</center>
My “prepared” mind searched for answers, leading me down varied learning paths. Tapping the power of networks, I dropped a note to Chance News editor Bill Peterson. His quick web search surfaced a story from ''Nature News'' on research by Hans Herrmann, et. al. [http://www.nature.com/news/2004/040227/full/news040223-11.html Shattered eggs reveal secrets of explosions].  As described there, researchers have found power-law relationships for the fragments produced by shattering a pane of glass or breaking a solid object, such as a stone. Seems there is a science underpinning how things break and explode; potentially useful in Forensic reconstructions.
Bill also provided a link to [http://cran.r-project.org/web/packages/poweRlaw/vignettes/poweRlaw.pdf a vignette from CRAN] describing a maximum likelihood procedure for fitting a Power Law relationship. I am now learning my way through that.


A brief search for other estimates of the chance that an individual skilled forecaster could win the prize finds
Submitted by William Montante
*Jeff Bergen of DePaul University asserts [http://www.youtube.com/watch?v=O6Smkv11Mj4in this youtube video]  a 1 in 128 billion chance.
*Ezra Miller of Duke University [http://www.latimes.com/business/la-fi-buffett-basketball-bet-20140122,0,7653962.story#axzz2r88PpgCI is quoted as saying] a 1 in 1 billion chance.
Neither source explains how these chances were calculated.


Submitted by David Aldous
----

Latest revision as of 20:58, 17 July 2019


Forsooth

Quotations

“We know that people tend to overestimate the frequency of well-publicized, spectacular events compared with more commonplace ones; this is a well-understood phenomenon in the literature of risk assessment and leads to the truism that when statistics plays folklore, folklore always wins in a rout.”

-- Donald Kennedy (former president of Stanford University), Academic Duty, Harvard University Press, 1997, p.17

"Using scientific language and measurement doesn’t prevent a researcher from conducting flawed experiments and drawing wrong conclusions — especially when they confirm preconceptions."

-- Blaise Agüera y Arcas, Margaret Mitchell and Alexander Todoorov, quoted in: The racist history behind facial recognition, New York Times, 10 July 2019

In progress

What if the Placebo Effect Isn’t a Trick?
by Gary Greenberg, New York Times Magazine, 7 November 2018

The Problems With Risk Assessment Tools
by Chelsea Barabas, Karthik Dinakar and Colin Doyle, New York Times, 17 July 2019

Hurricane Maria deaths

Laura Kapitula sent the following to the Isolated Statisticians e-mail list:

[Why counting casualties after a hurricane is so hard]
by Jo Craven McGinty, Wall Street Journal, 7 September 2018

The article is subtitled: Indirect deaths—such as those caused by gaps in medication—can occur months after a storm, complicating tallies

Laura noted that

Did 4,645 people die in Hurricane Maria? Nope.
by Glenn Kessler, Washington Post, 1 June 2018

The source of the 4645 figure is a NEJM article. Point estimate, the 95% confidence interval ran from 793 to 8498.

President Trump has asserted that the actual number is 6 to 18. The Post article notes that Puerto Rican official had asked researchers at George Washington University to do an estimate of the death toll. That work is not complete. George Washington University study

We sttill don’t know how many people died because of Katrina
by Carl Bialik, FiveThirtyEight, 26 August 2015

These 3 Hurricane Misconceptions Can Be Dangerous. Scientists Want to Clear Them Up.
Misinterpretations of the “Cone of Uncertainty” in Florida during the 2004 Hurricane Season
Definition of the NHC Track Forecast Cone

Remember when a glass of wine a day was good for you? Here's why that changed. Popular Science, 10 September 2018

Googling the news
Economist, 1 September 2018

We sat in on an internal Google meeting where they talked about changing the search algorithm — here's what we learned

Reading , Writing and Risk Literacy

[1]

Today is the deadliest day of the year for car wrecks in the U.S.

Some math doodles

<math>P \left({A_1 \cup A_2}\right) = P\left({A_1}\right) + P\left({A_2}\right) -P \left({A_1 \cap A_2}\right)</math>

<math>P(E) = {n \choose k} p^k (1-p)^{ n-k}</math>

<math>\hat{p}(H|H)</math>

<math>\hat{p}(H|HH)</math>

Accidental insights

My collective understanding of Power Laws would fit beneath the shallow end of the long tail. Curiosity, however, easily fills the fat end. I long have been intrigued by the concept and the surprisingly common appearance of power laws in varied natural, social and organizational dynamics. But, am I just seeing a statistical novelty or is there meaning and utility in Power Law relationships? Here’s a case in point.

While carrying a pair of 10 lb. hand weights one, by chance, slipped from my grasp and fell onto a piece of ceramic tile I had left on the carpeted floor. The fractured tile was inconsequential, meant for the trash.

BrokenTile.jpg

As I stared, slightly annoyed, at the mess, a favorite maxim of the Greek philosopher, Epictetus, came to mind: “On the occasion of every accident that befalls you, turn to yourself and ask what power you have to put it to use.” Could this array of large and small polygons form a Power Law? With curiosity piqued, I collected all the fragments and measured the area of each piece.

Piece Sq. Inches % of Total
1 43.25 31.9%
2 35.25 26.0%
3 23.25 17.2%
4 14.10 10.4%
5 7.10 5.2%
6 4.70 3.5%
7 3.60 2.7%
8 3.03 2.2%
9 0.66 0.5%
10 0.61 0.5%
Montante plot1.png

The data and plot look like a Power Law distribution. The first plot is an exponential fit of percent total area. The second plot is same data on a log normal format. Clue: Ok, data fits a straight line. I found myself again in the shallow end of the knowledge curve. Does the data reflect a Power Law or something else, and if it does what does it reflect? What insights can I gain from this accident? Favorite maxims of Epictetus and Pasteur echoed in my head: “On the occasion of every accident that befalls you, remember to turn to yourself and inquire what power you have to turn it to use” and “Chance favors only the prepared mind.”

Montante plot2.png

My “prepared” mind searched for answers, leading me down varied learning paths. Tapping the power of networks, I dropped a note to Chance News editor Bill Peterson. His quick web search surfaced a story from Nature News on research by Hans Herrmann, et. al. Shattered eggs reveal secrets of explosions. As described there, researchers have found power-law relationships for the fragments produced by shattering a pane of glass or breaking a solid object, such as a stone. Seems there is a science underpinning how things break and explode; potentially useful in Forensic reconstructions. Bill also provided a link to a vignette from CRAN describing a maximum likelihood procedure for fitting a Power Law relationship. I am now learning my way through that.

Submitted by William Montante

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