Archive for lottery

Brad Carlin’s take on Warren Buffett’s $1 billion offer [guest post]

Posted in Kids, Statistics, University life with tags , , , , on January 25, 2014 by xi'an

[This is a guest post from Brad Carlin, prepared for a news channel that never called back. If, like me, you have no idea of what March Madness means or of the meaning of brackets and seeds in this context, read the insert below first. Brad kindly added it upon my request...]

In mid-March every year, the national champion in US college basketball is decided through a 6-round tournament called “March Madness”: 64 teams are seeded into the tournament based on their perceived strengths, with 1 seeds going to top top 4 teams and 16 seeds going to weakest teams. A couple years ago the tourney added a partial 7-round, with just 4 more teams added (winning one of those those 4 “play-in” games merely gives you the right to be one of the 64 — and probably to lose in the next round to a top seeded team).

Anyway, every year Americans fill out “their brackets”, which means once the tourney is announced, it’s very very common to participate in a pool (often at your office) where everybody throws in some small amount of money ($10?) and submits a prediction of how *all* 67 games are going to come out, with the guy whose sheet most accurately predicts the actual outcome winning the money. Pool sheets are typically scored in some systematic way, the most common being 1 point for the first round (or play-in game, if included; many poolmasters just give you the play-in game winners for free), 2 for the second round, 4 for the 3rd, 8 for the 4th, and so on (the exponential award system reflecting how hard it is to keep getting the predictions right as the tourney progresses). Office pools have become so popular that it’s estimated that March Madness is now the most wagered-upon event in the world, passing even the Super Bowl (American pro football championship). Strictly speaking, as a form of gambling office pools are illegal in most states, but the cops generally look the other way provided the poolmaster isn’t taking a cut of the pot as a fee; all moneys collected must be distributed in prizes.

There are websites to tell you how best to fill out your poolsheet; poologic is one that I’ve been involved with, at least in some small way; see my name on a paper on the bottom of the program’s info page. My university has allowed (nay, encouraged) me to speak to reporters about March Madness when it comes around every year, and this year the big story is that Omaha billionaire Warren Buffett is underwriting an ad campaign by Quicken Loans that will pay you $1B if you get the whole bracket completely right — all 67 games, even the play-in games. The odds against this are astronomical (even with 10 million players), so my initial reaction was it’s not a very interesting problem; as Jeff Rosenthal might say, you’re more likely to be struck by lightning on your way to handing in your pool sheet than you are to win the prize. But the part I did find interesting is speculating about how much Mr. Buffett (the 2nd or 3rd richest man in the nation, and a very shrewd investor) charged Quicken for his agreement to underwrite (ie pay off the best if by some miracle somebody actually won). The blog post below suggests he probably only needed to charge about $78k to make the bet “fair”, but in fact he probably charged much, much more than this — a fee he will in all likelihood keep and smile as he walks to the bank.

Warren Buffett, Quicken Loans offer $1 billion for a perfect bracket during March Madness 2014

Hey, want to win $1 billion? Well, of course you do! All you have to do is fill out a perfect bracket for the NCAA men’s basketball tournament otherwise known as March Madness, and billionaire businessman Warren Buffett and Quicken Loans will send you $25 million a year for 40 years.

No big deal, you say? Think again: there are roughly 148 quintillion ways to fill out your bracket, the result of having to make 67 picks (the 63 games in the tournament proper plus the 4 “play-in” games).

Even though some of these picks are easy to make (say, that a 16 seed will not defeat a 1 seed, since no 16 has ever won a game in the men’s tournament), the odds of successfully completing a perfect bracket are astronomically small — somewhere between 1 in a billion and 1 in 128 billion.

So you’re saying there’s a chance? Yes, but a very slim one at best.

Brad Carlin, Ph.D., is a professor of biostatistics in the School of Public Health at the University of Minnesota and has been following this story since it unfolded. “This is a fascinating case, especially for me as a statistician long interested in NCAA tournaments and wagering. Mr. Buffett is essentially acting as the insurance company for Quicken, and it’s fun to speculate on how much he charged Quicken for this ‘coverage’.”

Rules of the contest specify only the first 10 million participants who register will be allowed to submit brackets. Assuming independent players each submitting one entry, standard probability calculations reveal the probability that at least one of them will achieve perfection. Multiplying by $1 billion produces the “premium” Mr. Buffett would need to charge Quicken to exactly cover the risk.

However, this premium varies widely depending on how hard you assume a perfect bracket is to achieve. Prof. Ezra Miller of Duke University suggests a skilled player would have a 1 in a billion chance of perfection; there is about a 1% chance that at least one of 10,000,000 such players would end up perfect. This leads to a premium of $1B x .01 = $10,000,000, a tidy sum indeed. However, Prof. Jay Berger of DePaul University thinks even a skilled player would have at best a 1 in 128 billion chance of perfection. The chance of at least one of 10,000,000 such players beating Mr. Buffett is only 0.0078%, leading to an insurance premium of just $78,000 – a relative pittance. Neither Mr. Buffett nor Quicken will reveal the actual premium, but it seems likely to be much closer to $10M than $78K; Quicken is donating $3M for home loans and other charitable causes as part of the promotion regardless of whether anyone hits perfection or not.

Carlin adds Mr. Buffett’s risk is probably even lower than this due to the lack of independence among players. “The best way to beat Mr. Buffett here would be to enter the 10,000,000 most likely pool sheets, which can be computed from point spreads and team computer ratings once the brackets come out.” But instead, Carlin predicts the public will likely do as they always do: overvalue “conventional wisdom” and overbet the favorites (higher seeds) in the tournament, leading to many pool sheets that look largely – or perhaps even exactly – alike. “Positive dependence among the 10,000,000 entries makes it even easier for Warren to hang onto his billion.”

Regardless of how they make their picks, odds, participants are going to have to overcome monumental odds to rake in the prize. For those who decide to get involved anyway, Carlin counsels diversity: “Don’t be afraid to pick a few upsets; unlike most March Madness challenges, here the goal is perfection, not merely point accumulation. Take more than a few chances to make sure your sheet differs from the other 9,999,999.”

For more from Carlin on picking March Madness winners, see this past Health Talk blog post.

Mr Meyrowitz’s glasses

Posted in Statistics, University life with tags , , , , , , , on October 23, 2011 by xi'an

Today, I found a site entitled Mr Meyrowitz’s Class that links to my first post on coincidences in lotteries as an example of “fatal error”. This seems to be part of a student’s assignment, apparently for the CollegeBoard programme, with 10 minutes allocated to students to find my “fatal error with decimals and probabilities”… As there is no hint, I wonder where my fatal error stands: I could not find it after those 10 minutes of intense searching and recomputing. Maybe Mr Meyrowitz actually needs new glasses to spot the difference between a 1‰ chance and a 1% chance… (Which actually misled a few other readers of the post.)

Question 6) in this assignment also sounds very much inspired from another of my posts on coincidences in lotteries [although not acknowledged in the assignment] since the question refers to the same original France Soir article in French. The question is however rather vague: “do you suspect him of cheating?” and it shows a lack of knowledge about French loto where cheating is [close to] impossible. It is certainly not recommended as an exercise for beginning students in probability or statistics. [Actually, in my opinion, the whole assignment is poor, being either imprecise, e.g question 7), useless, as for question 4) "Pick one topic that you understand very well and one that you do not understand well" (!), or plain wrong, as for question 2)...]

Another coincidence…

Posted in Mountains, pictures, Travel with tags , , , , , , , on September 9, 2011 by xi'an

After the coincidence of bumping into Marc Suchard in an Edinburgian Indian restaurant on Tuesday night, I faced another if much less pleasant coincidental event: for the third time in a row, my bag went missing on a Air France flight to Scotland… This happened first for the mixture meeting in 2009, costing me an attempt at Tower Ridge on Ben Nevis, then again when I took part in the colloquium celebrating Mike Titterington last May. Since having three independently lost luggages on three (or six) trips is very unlikely, there must be a reason to this pattern! Besides a conspiracy theory about the airline pushing me towards other companies because of my grumpy patronage or my unsuccessful requests for Irn Bru, possible reasons are late checking-ins (even though this does not apply in the last two cases) and use of a [small] backpack that is always turned into an “oversized” piece of luggage by the airline ground personal (does not apply to the first occurrence), but I do not carry particularly suspicious items, not even haggis on the way back… There must be a better reason than that!

another lottery coincidence

Posted in R, Statistics with tags , , , on August 30, 2011 by xi'an

Once again, meaningless figures are published about a man who won the French lottery (Le Loto) for the second time. The reported probability of the event is indeed one chance out of 363 (US) trillions (i.e., billions in the metric system. or 1012)… This number is simply the square of

{49 \choose 5}\times{10 \choose 1} = 19,068,840

which is the number of possible loto grids. Thus, the probability applies to the event “Mr so-&-so plays a winning grid of Le Loto on May 6, 1995 and a winning grid of Le Loto on July 27, 2011“. But this is not the event that occured: one of the bi-weekly winners of Le Loto won a second time and this was spotted by Le Loto spokepersons. If we take the specific winner for today’s draw, Mrs such-&-such, who played bi-weekly one single grid since the creation of Le Loto in 1976, i.e. about 3640 times, the probability that she won earlier is of the order of

1-\left(1-\frac{1}{{49\choose 5}\times{10\choose 1}}\right)^{3640}=2\cdot 10^{-4}.

There are thus two chances in 10 thousands to win again for a given (unigrid) winner, not much indeed, but no billion involved either. Now, this is also the probability that, for a given draw (like today’s draw), one of the 3640 previous winners wins again (assuming they all play only one grid,  play independently from each other, &tc.). Over a given year, i.e. over 104 draws, the probability that there is no second-time winner is thus approximately

\left(1-\frac{1}{2\cdot10^4}\right)^{104} = 0.98,

showing that within a year there is a 2% chance to find an earlier winner. Not so extreme, isn’t it?! Therefore, less bound to make the headlines…

Now, the above are rough and conservative calculations. The newspaper articles about the double winner report that the man is playing about 1000 euros a month (this is roughly the minimum wage!), representing the equivalent of 62 grids per draw (again I am simplifying to get the correct order of magnitude). If we repeat the above computations, assuming this man has played 62 grids per draw from the beginning of the game in 1976 till now, the probability that he wins again conditional on the fact that he won once is

1-\left(1-\frac{62}{{49 \choose 5}\times{10 \choose 1}}\right)^{3640} = 0.012,

a small but not impossible event. (And again, we consider the probability only for Mr so-&-so, while the event of interest does not.) (I wrote this post before Alex pointed out the four-time lottery winner in Texas, whose “luck” seems more related with the imperfections of the lottery process…)

I also stumbled on this bogus site providing the “probabilities” (based on the binomial distribution, nothing less!) for each digit in Le Loto, no need for further comments. (Even the society that runs Le Loto hints at such practices, by providing the number of consecutive draws a given number has not appeared, with the sole warning “N’oubliez jamais que le hasard ne se contrôle pas“, i.e. “Always keep in mind that chance cannot be controlled“…!)

Common ancestors

Posted in Statistics with tags , , , , on June 8, 2011 by xi'an

In conjunction with President Obama’s visit to Ireland two weeks ago and in particular to his ancestral Irish town, I happened to glance at his family tree and saw that he shared a common ancestor with George W. Bush. (They are 11th cousins, meaning that a 12th-order ancestor is common to both their  family trees.) This sounds at first amazing, but it is another occurrence of the (von Mises) birthday problem. (The fact that it is not that amazing is demonstrated by the simultaneous presence of [French!] ancestors of Dick Cheney in the same tree.) If we consider President Obama’s mother side, the probability that all of her 11th-order ancestors differ from all of George W. Bush’s 12th-order ancestors is

p=\dfrac{(M-2^{12})(M-2^{12}-1)\cdots(M-2^{12}-2^{11}+1)}{M(M-1)\cdots(M-2^{11}+1)}

where M denotes the whole population of potential ancestors at this period. If we consider all those ancestors as coming from the British Isles, then about 1650, the population was about 8 million. This would lead to a probability of p=0.35, i.e. there is a 65% chance that they share a 12th-order ancestor. If instead we consider the whole European population at that time (if only to include German and French ancestors to President Obama), M is about 100 million and the probability increases to p=0.92, so there is then an 8% probability for them to share an ancestor. Obviously, this rough calculation relies on simplifying assumptions, avoiding the issue of inbreeding which means that the potential 2¹¹ ancestors are in fact much less than 2¹¹, and the fact that their ancestors are necessarily emigrants, which reduces the value of M(This post appeared yesterday on the Statistics Forum.)

Another coincidence in lotteries

Posted in Statistics with tags , , , on January 10, 2011 by xi'an

Here is a lottery story that appeared in the LA Times (and in numerous other journals and news reports):

On the television show Lost, the character Hugo “Hurley” Reyes played the numbers 4, 8, 15, 16, 23 and 42 and ended up winning the $114-million jackpot. In real life, the lotto’s selections for its $355-million prize — 4, 8, 15, 25, 47 and the crucial “Mega ball,” 42 — included four of Hurley’s numbers. According to the Mega Millions website, which reported receiving “unprecedented traffic” after the drawing, 41,763 people matched those four numbers, earning $150 apiece.

Matching at least four numbers out of six is not that unlikely. Assuming there are 48 possible numbers in the lottery, and not bothering about specifics for this lottery like “Mega ball” and specific order, a basic probability computation for this match is

{6\choose 4}{44\choose 2}/{48\choose 6}=0.001156,

that is a 1‰ chance when taking this lottery and this sequence of numbers in isolation. This probability does not account for the parallel and subsequent runs of many other lotteries where many players will have undoubtedly used those “lucky” numbers, since they seem to have been around for several years, nor for the fact that other series have also proposed lottery numbers (just a few weeks ago my daughter was watching an episode of the series Cold Case which involved a lottery draw).

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