Probability and the Florida Lottery

The usual probability story in this context is something like the following. A New Jersey woman, Evelyn Adams, won the lottery twice within a span of four months raking in a total of 5.4 million dollars. She won the jackpot for the first time on October 23, 1985 in the Lotto 6/39 in which you pick 6 numbers out of 39. Then she won the jackpot in the new Lotto 6/42 on February 13, 1986. Lottery officials calculated the probability of this as roughly one in 17.1 trillion, which is probability that one preselected person won the lottery on two preselected dates.

When one realizes that (i) somebody won the October 23, 1985 lottery. (ii) We would have been equally impressed if this happened twice within a one year period. (100 twice weekly drawings) (iii) Many people who play the lottery buy more than one ticket. Taking these three things into account he probability ends up to be about is now about 1/200. If we take into account the number of states with lotteries. For more examples of things that aren’t as surprising as they seem look at http://www.math.duke.edu/~rtd/Talks/Emory.pdf.

A recent paper on the arXiv:1503.02902v1 by Rich Arratia, Skip Garibaldi, Lawrence Mower, and Philip B. Stark tells a different type of story. In Florida’s Play 4 game you pick a four digit number like 3782 and if all four digits match you win $5000. The fact that this event has probability 1/10000 and hence nets you 0.50 average, says either that (i) people can’t think or (ii) they have utility functions that value a large sum disproportionately more than the $1 you use to play the game.

Some people however are very good at winning this gamble. An individual that we will call LJ has won 57 times. Now that by itself is not proof of guilt. If he bought 570,000 tickets he would end up with about this many wins. However that seems a little unlikely. If he only bought 250,000 tickets the probability of 57 wins is 1.22 x 10-8. (Exercise for the reader.)

Arratia et al give a very nice calculation that shows something funny must be going on. Skipping the math, the bottom line is that if the 19 million people that live in Florida all sold their houses, and took the $175,000 in proceeds (this is the average house value) and bought lottery tickets (reinvesting the winnings) until they ran out of money, the probability that someone would win 57 times or more is 1 in a million.

How did LJ get so lucky? Well there are three common schemes. (i) A clerk can scratch the ticket with a pin revealing enough of the bar code to be able to scan it to see if it is a winner. (ii) Sometimes a customer will ask the clerk if the ticket was a winner. If so the clerk may lie about the ticket being a winner and keep the money himself. (iii) Sometimes the winner may be an illegal immigrant or owe child support or back taxes, and will sell the ticket to an aggregator who pays half price for it and later claims the prize. This is a good scheme for people who want to launder money.

It would be nice if I could tell you that probability helped catch a criminal but at least it wasn’t involved in a miscarriage of justice like Sally Clark experienced. She was convicted of murder based on the calculation that the odds were 73 million to 1 against two of her children dying of what is called cot death in the UK.

Two movies about Alan Turing

The Imitation Game (IG) is a great movie which has brought a lot of attention to Alan Turing but if you like it, you should also watch the 2013 film Codebreaker (CB), which can be streamed on Netflix. Remarkably these two films give almost disjoint accounts of his life. I guess at this point I should give a SPOILER ALERT that I am about to describe some pivotal events in his life, which are revealed in the movie. A couple of the revelations might spoil your enjoyment but you have been warned.

CB spends a fair amount of time on Turing’s work on computability. The movie even shows a copy of the 1928 article by David Hilbert’s paper “On Computable Numbers, with an Application to the Entscheidungsproblem (Decision Problem).” It goes into some detail describing what a Turing machine is. As many of you know, he proved that such a machine would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. The movie doesn’t go on to mention that Turing showed that the halting problem for Turing machines was undecidable, but that’s OK since Von Neumann acknowledged that the central concept of the modern computer was due to this paper. Not a bad result for an undergraduate at King’s College.

IG spends a lot of telling the story of Turing’s role in decrypting messages sent by the Enigma machine. At Bletchley Park, Turing built an electromechanical machine, the bombe, that could help break Enigma more effectively than the Polish bomba kryptologiczna, from which its name was derived.

The bombe searched for possible correct settings used for an Enigma message (i.e. rotor order, rotor settings and plugboard settings), using a suitable crib: a fragment of probable plaintext. For each possible setting of the rotors, which had of the order of 10 19 states, or 1022 for the four-rotor U-boat variant, the bombe performed a chain of logical deductions based on the crib, implemented electrically. It detected when a contradiction had occurred, and ruled out that setting, moving on to the next. A brute force search of such a large space was not practical. According to the movie, a breakthrough came when they realized that the Germans sent out a weather report each day at 6AM that ended with the phrase Heil Hitler.

Both movies mention the fact that in 1941, Turing proposed marriage to Hut 8 co-worker Joan Clarke (played by Keira Knightley in IG), a fellow mathematician and cryptanalyst, but their engagement was short-lived. After admitting his homosexuality to his fiancée, who was reportedly “unfazed” by the revelation, Turing decided that he could not go through with the marriage. (In real life Joan Clark was somewhat less attractive.)

Despite some scenes of Turing running long distances (in CB if I recall correctly) neither movie mentions that while working at Bletchley, Turing, who was a talented long-distance runner, occasionally ran the 40 miles to London when he was needed for high-level meetings. In addition, he was capable of world-class marathon standards. Turing tried out for the 1948 British Olympic team. His time for the marathon was only 11 minutes slower than British silver medalist Thomas Richards’ Olympic race time of 2 hours 35 minutes.

Going back in time, a third key event in Turing’s life occurred at Sherborne School, which Turing entered in 1926 at the age of 13. At Sherborne, Turing formed an important friendship with fellow pupil Christopher Morcom, which provided inspiration in Turing’s future endeavours. However, the friendship was cut short by Morcom’s death in February 1930 from complications of bovine tuberculosis contracted after drinking infected cow’s milk some years previously. This event shattered Turing’s religious faith and he became an atheist

Neither movie has anything to say about his ground breaking paper on The Chemical Basis of Morphogensis published in 1952, which put forth his ideas about pattern formation in development. However, both movies cover the fact that in January 1952, Turing, then 39, started a relationship with Arnold Murray, a 19-year old unemployed man. A burglary brought the police to his house, the police discovered their relationship, and the fact that a valuable watch was missing was forgotten.

Homosexual acts were criminal in the UK at that time.  Turing was convicted and given a choice between imprisonment and probation, which would be conditional on his agreement to undergo hormonal treatment designed to reduce libido. He accepted the option of treatment via injections of stilboestrol, (CB shows you the bottle of tablets), a synthetic estrogen. The treatment rendered Turing impotent and caused gynaecomastia, growing female breasts.

On 8 June 1954, Turing’s housekeeper found him dead. He had died the previous day. A post-mortem examination established that the cause of death was cyanide poisoning. When his body was discovered, an apple lay half-eaten beside his bed, and although the apple was not tested for cyanide, it was speculated that this was the means by which a fatal dose was consumed.

CB spends more time on the impact of estrogen therapy than IG, which has one brief scene with Turing and Joan Clarke one year after his conviction, in which he shows tremors in his movements. CB makes the point that the hormones did more stop his sex drive they also affected his ability to think. This part of the story is told in IG through a conversation between Turing and policeman, which explains the title Imitation Game.  In “Computing machinery and intelligence,” Turing addressed the problem of artificial intelligence, and proposed an experiment which became known as the Turing test, an attempt to define a standard for a machine to be called “intelligent”. The idea was that a computer could be said to “think” if a human interrogator could not tell it apart, through conversation, from a human being.

The achievements listed above do not exhaust all the extraordinary things Turing did in his 41 years. IG portrays him as an overbearing individual who could not understand other people’s feelings. Today we would say he was on the autism spectrum. CB tells the story of a brilliant man who just happened to be gay. Independent of which of these (if either) is true, IG says in its closing moments that cracking the Enigma Code shortened the war by two years and SAVED 14 MILLION LIVES.

Given this and his impressive intellectual achievements, the decision to chemically castrate Turing, which caused his death was insane, as is the fact that it took until 2009 for the British Government to apologize. With an important decision on Gay Marriage looming in the Supreme Court, this is an important example to keep in mind.  When homophobes and bigots quote the Bible to justify that marriage is only allowed between one man and one woman, we should ask “What would Jesus do?”

For more on Turing you could buy the book Alan Turing: The Enigma written by Andrew Hodges and Douglas Hofstadter or visit www.turing.ord.uk/, maintained by Hodges.

The Joys of Flying are Without Number

The title is stolen from a cartoon. The sentence continues with a colon and then the number 0. Friday January 9, 2015 I had an AMS council meeting in San Antonio. The meeting was 2:30-10PM so I decided to fly in that morning and then out the next morning. The Monday before my flight, Delta emailed me through Orbitz to say that I was rebooked on a 7:10AM flight, which then entailed getting up at 5AM to get to the airport on time.

I got to a coldy and rainy San Antonio without incident, and except for having to pay $20 for two enchiladas and a beer at an almost deserted river walk eatery, all was well until about 4PM when I was bored and checked my email.

Hi RICHARD,

Unfortunately, US Airways 1983 to Charlotte Douglas (CLT) has been canceled. Please contact Orbitz at 800-656-4546 to reschedule your flight. Provide your record locator: AP1101013JY4VWS6 when you call.

I snuck out of the meeting and called Orbitz and called them. Each of the calls ended badly. The first one said “you called the wrong number.” I got transferred to another number and disconnected. On a second try, after four minutes on hold, I got someone who told me “that’s not your record locator” and I got disconnected. After I retrieved the booking record locator, I tried a third time. The person said she couldn’t help me but would transfer my call, and I was once again disconnected.

Since the flight was on USAir I called them. A helpful woman told me that I had been rebooked on an American flight that left at 8:50AM went through Dallas and got to RDU at 2:30PM. I thanked her profusely, went backed to the meeting. We finished up business at 6:30 had a couple of glasses of wine and dinner then a final discussion 8-9.

Tired but happy I went back to my room and watched some golf. After a while I decided I should check on my flight. I logged on to American, navigated to my flight and there it was: they had replaced my return on Saturday January 10 by one on Sunday January 11. This was upsetting because I didn’t have a hotel reservation or a change of clothes for the extra day. After a couple of calls, first to American and then to USAir, I had a flight leaving at 12:45 Saturday and arriving back to RDU at 6:53.

Though it was now 10PM I went down to the bar for a couple of glasses of wine to soothe my jangled nerves. I ran into SAMSI director Richard Smith who told me that San Antonio was expecting freezing rain, which is why all the flights Saturday morning were cancelled.

Saturday morning I slept in until 6:30. I opened the curtains to see my magnificent view of the Denny’s across the street and the freeway in the background. Very light rain and 35 degrees. Oh well, at least I didn’t have to set my alarm early and rush to get to the airport. I did some work, and some time later sitting in the San Antonio airport, I wrote this.

I am happy to say that the rest of the trip went well and I got home about 7:45. When I got to Dallas an checked my email, I got a note from the ever helpful Orbitz saying that my flight from Charlotte to RDU was on time. Not sure why I should care since the flight from San Antonio to Charlotte was cancelled.

The Viagra Standard: Redux

It seems fitting to open my new blog by going back to one of the first columns I wrote for the IMS Bulletin. At least I think that it was one of my firsts, the online archive doesn’t go back before 2010 and having changed computers since then, I can’t find the file.

The issue then as it is today is: What is good probability? Several of my recent papers have been rejected by referees who said: “this is interesting, but it isn’t hard.” Years ago I coined the phrase the Viagra standard for this – if it is not hard it is not good. The phrase never got into print because the editor thought it was a little risqué. I agree that there is something obscene here, but it is the policy not the phrase. Do we really want journals filled with papers that are hard but not interesting? Evidently we do, because that’s what we have got!

Rather than argue abstractly, let me talk about one concrete example. During the 2010-2011 academic year, David Sivakoff and I had almost weekly meetings with a group of students from the North Carolina School of Science and Math. We eventually wrote a paper with two of the students from the group, Sam Magura and Vichtyr Pong. The paper was inspired one published in the Proceedings of the National Academy of Science by Henry, Pralat and Zhang [108: 8605-8610].

People have opinions in the d-dimensional unit cube. Connections are broken at a rate proportional to their length and rewired randomly, or reconnected by a Metropolis-Hastings dynamic that always accepts shorter connection and accepts longer one with a probability equal to the ratio of the lengths. The key to our analysis was that the system had a reversible stationary distribution that is closely related to long range percolation, modulo the difference that the evolving graph has a fixed number of edges while the percolation has a random number.

There was also a second model invented by the students, which involved individuals who preferred to be connected to people who are popular. Again there was a reversible stationary distribution, which this time was related a graph generated by a particular instance of the configuration model. The paper proved some results about the two systems and did simulations to treat what we could not.

In December 2012 we submitted the paper to Electronic Journal of Probability. The referee took seven months and decided: “My main concern is that too much of it is non-rigorous or is not sufficiently clearly stated to be rigorous. That said, the models discussed are interesting and there is some potential here.” We fixed the problems by using coupling to tighten up the connection between the evolving graphs with a fixed number of edges and the associated models that did not. We resubmitted the paper and heard nothing for five months. We wrote to the editor and we soon had a short report “Most of the comments I made in my review of a previous version of this paper have been addressed.” But the final decision was “The models are interesting as well as some of the results but the paper is perhaps not quite deep enough to merit publication in in EJP.”

Having lost more than a year at EJP, we turned to Journal of Applied Probability. Four months later we got a report from an expert in random graphs, who cited two of his own papers, and told us “I was disappointed reading the paper, since the authors proposed three interesting models but the analysis never went beyond what was easy to get and was already known.” It is frustrating to get absolutely no credit for making the connection and solving the problem. Yes we did use results about long range percolation, and Bollobas-Janson-Riordan, but these are hardly things everyone knows.

Perhaps the most depressing thing is that I looked at the list of papers to appear in JAP. Is our work really worse than what they print. The bottom line, boys and girls, is that if you want to get your work published, you should work in area that only ten people care about, and be friendly to them.