Some Quickie Probability

 Probability distributions via plus Magazine and 'Maths In a Minute':

https://plus.maths.org/content/what-are-probability-distributions

Statistical Illusion

 Today, from Futility Closet:

https://www.futilitycloset.com/2021/12/08/balance-8/

Terry Tao Tackles Probability and Risk

 Proposing a unit of measurement of risk:

https://terrytao.wordpress.com/2021/10/03/nines-of-safety-a-proposed-unit-of-measurement-of-risk/

Probability and Randomness via Twitter

 Oh, just another interesting Twitter thread:

https://twitter.com/svpino/status/1437793433255620633

Funny Dice....

 Funny dice (and probabilities) via John Cook (…and Donald Knuth… and, Garrison Keillor):

https://www.johndcook.com/blog/2021/08/16/lake-wobegon-dice/

Ponder This.... (a Twitter thread puzzle)

 Recent non-intuitive puzzle on Twitter from Joel David Hamkins:

https://twitter.com/JDHamkins/status/1407605175154532353

The Probability of....

 Interesting probability thread from David Butler this morning on Twitter:

https://twitter.com/DavidKButlerUoA/status/1377704121424224258

Bayesian Probability In the Day of Covid-19 (via NY Times)

Siobhan Roberts mathplains Bayesian analysis in the age of covid for NY Times readers:

https://www.nytimes.com/2020/08/04/science/coronavirus-bayes-statistics-math.html?smid=tw-share

Bye Bye Bayes…

                                           

One tidbit I found especially interesting/surprising (and had never heard about) from David Spiegelhalter’s current “The Art of Statistics” volume, is that Bayes' Theorem or Bayesian inference/reasoning has been banned in British courts… yeah, you read that right Nate Silver, BANNED in court! ;)  As if statistics weren’t confused and misused enough by lawyers, one of the most common and vaunted modern approaches to probability isn’t even allowed, and this decision goes back at least 8 years.

Here are a few of the links that talk about the decision:

https://www.theguardian.com/law/2011/oct/02/formula-justice-bayes-theorem-miscarriage

https://blog.richmond.edu/physicsbunn/2013/02/28/bayes-banned-in-britain/

https://telescoper.wordpress.com/2013/02/28/bayes-in-the-dock-again/

If any of our friends from across the pond can tell us more about how much controversy or debate this prohibition has produced in UK (or is it 'settled law' at this point?) I’d be curious to hear.

Insure THIS!

I found Chapter 14 (on “Weird Insurance”) of Ben Orlin’s delightful volume “Math With Bad Drawings” to be one of the most fun (of so many). It falls within an overall section on probability, and describes several non-standard insurances I’d never given much thought to. One of them he calls, “Oh No, My Employees All Won the Lottery” Insurance. We’ve all read stories in recent years of employees at a workplace banding together to purchase many tickets in some multi-million dollar lottery and then if one person wins, splitting the proceeds… and, then turning in their job resignations en masse.  As Ben puts it, it’s “like some kind of natural disaster” for the manager/proprietor of such a business. Oy.

Ben goes on to report that “In the 1990s, over a thousand British businesses invested in the National Lottery Contingency Scheme. If your staff won the jackpot, insurance payouts would help to cover recruitment, retraining, revenue losses, and temporary labor.”

Ben also points out that the employer could potentially self-insure by simply joining the employee pool and being one of the winner recipients (of course that assumes the employees even let the boss know what they're up to: "hey Boss, we're all joining together to win the lottery and quit this blankety-blank, friggin' workplace!").

…And if that’s not weird enough for you, shortly thereafter comes a section on “Alien-Abduction Insurance” :)  [...where Ben reports that a British company that sold 37,000 such plans hasn’t made a single payout yet (go figure!), and one of the managers involved says, “I’ve never been afraid of parting the feeble-minded from their cash”... so much for British understatement.]
Yeah, insurance is an interesting, creative business (selling you something you hope to never have to use).

Anyway, Ben’s book is a chuckle-a-minute, so if you want to insure against boredom, get it!

[The sections on probability and statistics constitute close to half the book and are especially fantastic, but the entire volume is a joy ride!]

Dueling Mathematicians...

First, in the Road-from-Ridicule-to-Nobel-Prize Dept.:

1)  In yesterday’s #BigMathOff competition Edmund Harriss linked to an older story I didn’t recall hearing, but found fascinating… that of Israeli scientist Daniel Shechtman ridiculed/mocked (even losing his job!) for discovering quasicrystals with never-repeating patterns, only to later receive the Nobel Prize for the same finding (after no less than Linus Pauling had said, “There is no such thing as quasicrystals, only quasi-scientists”):

https://www.reuters.com/article/us-nobel-chemistry/ridiculed-crystal-work-wins-nobel-for-israeli-idUSTRE7941EP20111006

The story is a great lesson in the uncertainty of science, and even a cautionary tale of the occasional difficulty in distinguishing science from crackpottery.

As many or most of you know, in recent years, there has been much attention given to possible links between quasicrystals and prime numbers or the Riemann Hypothesis:

https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/

https://golem.ph.utexas.edu/category/2013/06/quasicrystals_and_the_riemann.html

-------------------------------

And from the Surprises-In-Probability Dept.:

2)  Harriss’s competition in this round of the ‘Math-off’ is Colin Wright whose exposition of some full-deck card-play is wonderfully entertaining, especially for those who like playing with probabilities. It is the sort of example that can be enjoyed by both young people and adults, with a result that will likely seem counterintuitive at first, making it all the more enjoyable. Warning: it can be a bit addictive!

Kudos to Aperiodical's Christian Lawson-Perfect for organizing this 1st Annual ;) Big Internet Math-Off. The above Harriss-Wright square-off, by the way, is match 4 of round 1 (and you still have time to vote for your fave). Much fun yet to come.

Perception of Streaks

H/T to Dan Goldstein (on Twitter) for pointing out an Interesting plotting of (likely and VERY likely) “streak” probabilities:

http://www.decisionsciencenews.com/2017/10/25/long-need-flip-coin-see-streak/

The author looks mostly at "coin-tossing"-like scenarios, but notes such analysis potentially relates back to other theoretical discussions (such as "hot hand" observations). One can imagine a lot of other ways to play with similar data (and the author presents R code for doing such).

Update on Poker-playing Robot

A couple of weeks ago I mentioned that AI programs were now taking on poker, and a nice followups to that storyline appear this week:

a) https://www.technologyreview.com/s/603385/why-poker-is-a-big-deal-for-artificial-intelligence/

b) http://www.theverge.com/2017/1/25/14358246/ai-poker-tournament-cmu-libratus-vs-human-losing

In an ongoing Texas hold’em tournament a poker-playing robot named “Libratus” is so far up by almost $800,000 against its human competitors.

From the the first article: 

“Poker requires reasoning and intelligence that has proven difficult for machines to imitate. It is fundamentally different from checkers, chess, or Go, because an opponent’s hand remains hidden from view during play. In games of ‘imperfect information,’ it is enormously complicated to figure out the ideal strategy given every possible approach your opponent may be taking.”Further it is noted that “...an AI player has to randomize its actions so as to make opponents uncertain when it is bluffing.”

And from the 2nd article:

"One of the things Libratus does well is bluff..."Mastering the art of the bluff requires AI that can calculate risk and reward in real time without having perfect information about what its opponent can do in return. It implies the system does more than simply play a perfectly safe game where it only grinds out wins when it has the stronger hand."

Libratus is specifically programmed (as I understand it) to be skilled at just one specific poker game, so even if it wins this tournament (and it looks like it will, as it seems to be getting stronger over time), it doesn’t mean that bots are on the verge of taking over all professional poker… or, at least not yet. Of course the real congratulations go, not to the bot, but to the clever humans (in this case from Carnegie-Mellon) programming it.

==> ADDENDUM (1/31):  well, well, guess who won that tournament:

https://www.newscientist.com/article/2119815-ai-just-won-a-poker-tournament-against-professional-players/

Winning At Blackjack... and more

To start the week, this delightful (19-min.) Planet Money podcast on Ed Thorp, starting with his blackjack-beating escapades (h/t to Francis Su for this one):

http://one.npr.org/?sharedMediaId=510810966:510815770

Try to find time for it (all the more so if you don't already know Thorp's story.)

via WikimediaCommons

Mellifluous...

Yeah, sure, James Grime, Vi Hart, Matt Parker, are good, buuuuuuut...

is there any more mesmerizing voice for presenting math videos than Tadashi Tokieda!? The Perry Como of math perhaps... ;-) 
His latest for Numberphile here, on non-transitive dice:

And his other great vids here:

https://www.youtube.com/playlist?list=PLt5AfwLFPxWI9eDSJREzp1wvOJsjt23H_

(Non-transitive dice, by the way, come in different forms, some of which are available here:
https://mathsgear.co.uk/collections/dice )

Blast From the Past... the Quincunx

via Svjo/WikimediaCommons

Almost six years ago I posted about one of my earliest memories of something mathy that mesmerized me. I was interested in math and numbers at a young age, but my first view of a "quincunx" or "Galton board (or box)" was something that touched me at a deeper level.

So just a little nostalgia today as I re-run most of that earlier post below:

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Harking back to my childhood today....

When I was a youngster visiting a science museum in my home state what most fascinated me was not the dinosaur displays, or fossils, or insect collections, nor more whizbang exhibits, but a simple large display known as a "Galton Box" (after the 1889 inventor of the first one), or known by some as a "quincunx," (...okay, so I was an odd kid).

Most of you are likely familiar with these enclosed contraptions in which balls drop from a central point at the top onto a symmetrical pegboard where they bounce around until finally falling into columns at the bottom... the majority of balls dropping, by sheer chance, somewhere in the middle columns, and fewer balls bouncing around in a manner depositing them to the outer end columns.

On the glass pane enclosing the balls and peg-grid would be drawn the 'normal' or 'Gaussian' distribution (or 'Bell curve') so central to mathematics/probability, and lo-and-behold, once all the hundreds of balls had been released they would, in the columns below, take on the shape of that normal curve, via of course the 'laws' of sheer chance, not due to any mechanical manipulation. Even as a child, not really understanding much about normal distributions, nor math/probability more generally, somehow that demonstration was very powerful to me; like a magic, unseen hand guiding the fate of those individual spheres ---  each one taking a rather random, unpredictable journey, yet the end result being highly predictable and little-changing. Even as a youngster I sensed there was something profound in that. Some kids today construct Galton Boxes or quincunxes for science fair projects. Hooray!, for still to this day I love these apparatuses and their magical outcomes (...the rest of you can go gawk at dinosaur models).

************************************

Here's an animated version of a Galton Board from the Web:

http://www.mathsisfun.com/data/quincunx.html

And here's some chap you may recognize ;-) demonstrating a real-life miniature one:



...and finally this one, made from Legos, may give you some ideas if you wish to construct your own:
https://www.youtube.com/watch?v=aGiKdJ2npc4

Of Averages, Standard Deviations, Confidence Intervals, Probability

Worth noting perhaps that "Statistician" is one of the fastest-growing occupations in the American workplace:
http://thisisstatistics.org/statistician-projected-as-top-10-fastest-growing-job/

Some stat-related pieces catching my attention recently...

1)  Starting with the simplest, Todd Rose looks at learning and "averages" in this interview from NPR (...none of us are "average"):
http://tinyurl.com/jeryann

2)  More technical is Chris Harrow's look at standard deviations and sample size for his secondary math classroom:
https://casmusings.wordpress.com/2016/02/16/straightening-standard-deviations/

3)  Meanwhile, Deborah Mayo questions Nate Silver's explication of confidence intervals (includes interesting comment thread):
http://errorstatistics.com/2016/02/12/rubbing-off-uncertainty-confidence-and-nate-silver/

4)  And just this morning comes a long post from Jim Propp on probabilities involved in lottery values (and hat detection ;-):
https://mathenchant.wordpress.com/2016/02/16/when-not-to-expect-what-youre-expecting-3/

If stats aren't your thing then skip all these, but really, everyone should be acquainting themselves more with statistics.

The Improbability of Knowing Probability

via Gerald G/WikimediaCommons

On Oct. 21 Andrew Gelman asked on his blog, "What's the probability that Daniel Murphy hits a home run tonight?" (in a record-setting 6th straight playoff game):
http://andrewgelman.com/2015/10/21/whats-the-probability-that-daniel-murphy-hits-a-home-run-tonight/

He posted the answer as 20% and then, at the coaxing of some commenters, lowered it to 15%.
Then... later that evening, he raised the probability to 1... because of course Murphy (of the New York Mets) did just that, hit a home run in the 8th inning (playing against the Chicago Cubs, surely a major factor ;-)

And so, in a matter of hours the "probability" of something went from 20% to 15% to 100%... a nice demonstration of why, given human complexity, "probability" is often a near-meaningless concept when it comes to individual behavior and events.

Flippin' Probabilities

I suspect we all get a kick from mathematical problems that lead us merrily down wrong intuitive paths; statistics is often a source for such misdirecting problems. Mike Lawler pointed to a good one over the weekend. The problem arises in an old Peter Donnelly TEDTalk (below), starting at about the 4-minute mark (but his entire 22-minute talk is definitely worth a listen). The problem is simply this:

When flipping a fair coin repeatedly (heads/tails), which 3-part sequence is more likely to appear first:  HTH or HTT?
i.e., In a running sequence, is HTH likely to show up before HTT, or HTT before HTH, or, over many trials, are the probabilities equal?
.
.
.
.
.

MANY people jump to the conclusion that the two possibilities are equally likely, no doubt thinking in terms of the true equal probability of each single flip being an 'H' or a 'T.'
BUT NO, the somewhat surprising answer is that the HTT sequence is more likely to occur first, and the explanation Donnelly gives (again starting at ~4-min. point), has to do with the 'clumping' of overlapping 3-part chunks:



It may be worth noting that one could identify 3-part sequences that DO have equal probability of occurrence, as well as additional sequences, which like the above two, have unequal probabilities.

Donnelly doesn't really expound too fully on the explanation for the outcome, so you may wish to follow along Mike Lawler's videos (linked to above), or read up on "Penney's Game" (named after creator Walter Penney) which relates to all this:
http://en.wikipedia.org/wiki/Penney%27s_game

The always-entertaining "Scam School" once did a 15-min. episode on Penney's Game here:

Hmmm… Probability Puzzles

 I just recently discovered this blog which appears completely devoted to probability puzzles... making me feel just a tad like Homer Simpson… IF you substitute the phrase "probability puzzles" for "beer" ;-) :