Sandbox: Difference between revisions

From ChanceWiki
Jump to navigation Jump to search
 
(415 intermediate revisions by the same user not shown)
Line 1: Line 1:
==Lightning the the lottery==
[http://www.ctvnews.ca/canada/man-who-survived-lightning-strike-wins-1m-jackpot-with-co-worker-1.2478542 Man who survived lightning strike wins $1M jackpot with co-worker]<br>
CTV News (Canada), 20 July 2015


A Canadian man who survived a lightning strike at age 14 has just won a $1 million prize in the Atlantic Lotto, a 6/49 lottery game. Adding to the coincidence, his daughter is also a lightning survivor.  By combining the 1 in 13,983,816 chance of winning the Lotto with the reported 1 in a million risk of being stuck by lightning, a local math professor came up with the following "By assuming that these events happened independently … so probability of lotto … times another probability of lightning – since there are two people that got hit by lightning – we get approximately 1 in 2.6 trillion."


The story is amusing in light of the often-heard claim that one's chances of winning the lottery are the same as being stuck by lightning, a comparison of dubious utility. A quick search yields the following figures. According to the National Weather Service, there have been 24 US lightning deaths so far in 2015 (since 2010, the annual totals have been in the twenties). The accompanying map shows that one of the 2015 victims was from Texas; on the other hand the Winner's Gallery from the Texas Lottery displays 40 photos of happy winners. This of course includes some minor games in addition to the large jackpots that people might typically associate with the phrase "winning the lottery."  For more on this see the third discussion question below.
==Forsooth==


'''Discussion'''
==Quotations==
#Comment on the independence assumption in Canadian calculation.  (It may be instructive to look at the table of locations and activities for the US data on the Weather Service web site.)
“We know that people tend to overestimate the frequency of well-publicized, spectacular
#The 1 in a million figure given for lightning risk is presumably an annual figure.  But the man in the story was stuck as a teenager. How does this affect the calculation?
events compared with more commonplace ones; this is a well-understood phenomenon in
#See the blog post [https://talkingaboutnumbers.wordpress.com/2011/02/28/how-many-lottery-winners-are-there-in-a-year/ How many lottery winners are there in a year?] by Dan Ma for a more careful attempt to count lottery winners. Ma reports that data are not easy to come by.  Looking at  the 26-year period from the introduction of the California lottery until 2011, he counts 257 prizes of $1 million or more. This gives average of 10 per year.  From this, he estimates that the total for the US is likely in the hundreds. Comment on the comparison with lightning strikes. 
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>


Submitted by Bill Peterson
----


==More on the hot hand==
"Using scientific language and measurement doesn’t prevent a researcher from conducting flawed experiments and drawing wrong conclusions — especially when they confirm preconceptions."
In Chance News 105, the last item was titled [https://www.causeweb.org/wiki/chance/index.php/Chance_News_105#Does_selection_bias_explain_the_.22hot_hand.22.3F Does Selection bias explain the hot hand?].  It described how in their July 6 article, Miller and Sanjurjo assert that a way to determine the probability of a heads following a heads in a fixed sequence, you may calculate the proportion of times a head is followed by a head for each possible sequence and then compute the average proportion, giving each sequence an equal weighting on the grounds that each possible sequence is equally likely to occur.  I agree that each possible sequence is equally likely to occur.  But I assert that it is illegitimate to weight each sequence equally because some sequences have more chances for a head to follow a second head than others.


Let us assume, as Miller and Sanjurjo do, that we are considering the 14 possible sequences of four flips containing at least one head in the first three flips.  A head is followed by another head in only one of the six sequences (see below) that contain only one head that could be followed by another, making the probability of a head being followed by another 1/6 for this set of six sequences.
<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>


:{| class="wikitable" style="text-align:center"
==In progress==
|-
[https://www.nytimes.com/2018/11/07/magazine/placebo-effect-medicine.html What if the Placebo Effect Isn’t a Trick?]<br>
| TTHT  || Heads follows heads 0 time.
by Gary Greenberg, ''New York Times Magazine'', 7 November 2018
|-
| THTT  || Heads follows heads 0 times
|-
| HTTT  || Heads follows heads 0 times
|-
| TTHH  || Heads follows heads 1 time
|-
| THTH  || Heads follows heads 0 times
|-
| HTTH  || Heads follows heads 0 times
|}


A head is followed by another head six times in the six sequences (see below) that contain two heads that could be followed by another head, making the probability of a head being followed by another 6/12 = 1/2 for this set of six sequences.
[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


:{| class="wikitable" style="text-align:center"
==Hurricane Maria deaths==
|-
Laura Kapitula sent the following to the Isolated Statisticians e-mail list:
| THHT  || Heads follows heads 1 time
|-
| HTHT  || Heads follows heads 0 times
|-
| HHTT  || Heads follows heads 1 time
|-
| THHH  || Heads follows heads 2 times
|-
| HTHH  || Heads follows heads 1 time
|-
| HHTH  || Heads follows heads 1 time
|}


A head is followed by another head five times in the six sequences (see below) that contain three heads that could be followed by another head, making the probability of a head being followed by another 5/6 this set of two sequences.
:[Why counting casualties after a hurricane is so hard]<br>
:by Jo Craven McGinty, Wall Street Journal, 7 September 2018


:{| class="wikitable" style="text-align:center"
The article is subtitled: Indirect deaths—such as those caused by gaps in medication—can occur months after a storm, complicating tallies
|-
| HHHT  || Heads follows heads 2 times
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>
| HHHH || Heads follows heads 3 times
:by Glenn Kessler, ''Washington Post'', 1 June 2018
|}


An unweighted average of the 14 sequences gives
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.
<center>
[(6 &times; 1/6) + (6 &times; 1/2) + (2 &times; 5/6)] / 14 = [17/3] / 14 = 0.405,
</center>


which is what Miller and Sanjurjo report.
President Trump has asserted that the actual number is
A weighted average of the 14 sequences gives
[https://twitter.com/realDonaldTrump/status/1040217897703026689 6 to 18].
<center>
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.
[(1)(6 &times; 1/6) + (2)(6  &times; 1/2) + (3)(2 &times; 5/6)] / [(1&times;6) + (2 &times; 6) + (3 &times; 2)] <br>
[https://prstudy.publichealth.gwu.edu/ George Washington University study]
= [1 + 6 + 5] / [6 + 12 + 6] = 12/24 = 0.50.
</center>
Using an unweighted average instead of a weighted average is the pattern of reasoning underlying the statistical artifact known as Simpson’s paradox.  And as is the case with Simpson’s paradox, it leads to faulty conclusions about how the world works.  


Submitted by Jeff Eiseman, University of Massachusetts
:[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


===Comment===
----
<center>
[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>
{| class="wikitable" style="text-align:center"
[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]
! Sequence<br> of tosses !! Number of H <br> in first 3 tosses !! Number of H <br> followed by H !! Number of HH <br> in first 3 tosses!! Number of HH <br> followed by H
----
|-
[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.]
|  TTTT || 0 || 0  || 0 || 0
''Popular Science'', 10 September 2018
|-
----
| TTTH      || 0  || 0  || 0 || 0
[https://www.economist.com/united-states/2018/08/30/googling-the-news Googling the news]<br>
|-
''Economist'', 1 September 2018
| TTHT || 1  || 0  || 0 || 0
|-
| THTT || 1  || 0  || 0 || 0
|-
| HTTT || 1  || 0  || 0 || 0
|-
| TTHH || 1  || 1  || 0 || 0
|-
| THTH || 1  || 0  || 0 || 0
|-
| THHT || 2  || 1  || 1 || 0
|-
| HTTH || 1  || 0  || 0 || 0
|-
| HTHT || 2  || 0  || 0 || 0
|-
| HHTT || 2  || 1  || 1 || 0
|-
| THHH || 2  || 2  || 1 || 1
|-
| HTHH || 2  || 1  || 0 || 0
|-
| HHTH || 2  || 1  || 1 || 0
|-
| HHHT || 3  || 2 || 2 || 1
|-
| HHHH || 3  || 3 || 2 || 2
|-
! Total || 24  || 12  || 8 || 4
|}
</center>


==Percent change vs. percentage point change==
[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]
Mike Olinick sent a link to the following exchange:
----
[http://www.wyso.org/post/stats-stories-reading-writing-and-risk-literacy Reading , Writing and Risk Literacy]


:[http://www.nybooks.com/articles/archives/2015/jun/04/poor-college/ The poor in college]<br>
[http://www.riskliteracy.org/]
:by John S. Bowman, William Brigham, and Frank Robertson, ''The New York Review of Books'', 4 June 2015
-----
 
[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.]
These are responses to two earlier ''NYREV'' articles by Christopher Jenks:
:[http://www.nybooks.com/articles/archives/2015/apr/02/war-poverty-was-it-lost/ The war on poverty: Was it lost?], 2 April 2015
:[http://www.nybooks.com/articles/archives/2015/apr/23/did-we-lose-war-poverty-ii/ Did we lose the war on poverty?—II], 23 April 2015


==Some math doodles==
==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 \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|H)</math>


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

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

Retrieved from "https://www.causeweb.org/wiki/chance/index.php?title=Sandbox&oldid=22455"