Americans spend vast sums of money gambling on sporting events in everything from office pools to casino sports lines to illegal bookies to the internet. One of the biggest sporting events of the year is, of course, the Superbowl. Many people gamble directly on the outcome of the game, trying to beat the point spread. However, many casual gamblers enter a pool that is more like a lottery.

There are many variations of the basic football lottery pool, but I'll give the pool I've entered as an example. Players buy one or more squares on a 10X10 grid. Each square will represent the pair of last digits of the scores for each team. However, the numbers are not assigned to the grid axes until all the squares are sold. This is because some squares are much better than others and nobody in his right mind would buy certain squares. In the simplest pool, the player whose digits match the last digits of the final score wins. In some pools, prizes are awarded for the score at the end of each quarter as well as the final score, with the prize money escalating with each successive quarter. Another twist is to also give a small reward to the player who picks the reverse of the correct digits in each quarter. This last pool is the kind I am entered in. Thus, for my square, I have 8 ways of winning money.

I did an analysis of the likelihood of the digits, by quarter, based on the outcomes of all 39 previous Superbowl games. Some interesting facts emerge. As you'd expect, the digits 0, 3, and 7 are most likely, and some digits, for example 5 and 8, are downright terrible to be stuck with. There is a marked difference in the relative frequencies of the game winner's digits compared to those of the game loser's digits. For example, no losing Superbowl team has ever had a final digit of 5 at the end of any quarter, but 5 fares better than 8 for winning teams. For completeness sake, here is the combined relative frequency of all final digits, for winners and losers, in the scores.

The picture these graphs provide isn't complete. The sample size is on the small side. And for a fuller picture, what would be useful would be the joint likelihood of each of the 100 pairs of digits. (The winning and losing teams' digits are probably not independently distributed, making the joint likelihood more data-intensive to estimate.) Unfortunately, restricting the analysis to Superbowl games, there is nowhere near enough data to do this. Even looking at all recent playoff games wouldn't give a big enough statistical sample for that. So, until about Superbowl CC, the view here will have to suffice.

I pulled a 0 for the Steelers and a 6 for the Seahawks. The 0 is a great pick. The 6 is OK. I am rooting for the Steelers. The 0 would be even better if the Steelers end up being the losing team. But since I can also win some money on reverse digits, I'm happy with what I got. We'll see on Sunday...

**Update (2/07):** The digit frequency analysis has been updated for Superbowl XLI. The links above now reflect the more recent data. I've also worked up a digit pair frequency analysis. See the post for Superbowl XLI. I plan to keep updating the links for each successive Superbowl.

Who cares! The Bills aren't in it, so why do they even bother playing the game?

Posted by: Paul | February 03, 2006 at 11:28 PM

Why should playoff games be any different than regular season? You should have analyzed every game since the institution of the forward pass. That should have been a sufficient sample size.

BTW, I'm for the Steelers. I think Cower deserves it, even though he was way too loyal to Cordell Stewart. Plus Roethlisberger was

almosta Bill.Posted by: Chris | February 04, 2006 at 08:23 AM

PS How much did a square cost?

Posted by: | February 04, 2006 at 08:24 AM

Squares wer $10, so there's $1000 in prizes out there.

WIth regard to the sample size, I could've done as you say, but some people argue that SUperbowl games are different from playoffs which are different from regular season games. I'm a bit skeptical of that. We've had all kinds of Superbowl games, from total routs (Cowboys oer Bills, Bears over New England, etc) to the closest Superbowl of all time, the Bills 1 point loss to the Giants. Last year's game was interesting in that the first three quarters all ended in ties, but with different digits. In my pool, you win with the correct digits, but you also get a small prize for the correct reversed digits. Whoever had the (0,0), (7,7) and (4,4) squares won both ways last year. One thing is different in playoffs is that the game can't end in a tie. (I know ties were very rarem but they were still possible in the old days). Some people also think that some rule changes, partiuclarly ones involving pass defenses, have changed the game, and hence the scoring, over the last couple of decades. My analysis assumes that the statistics of Superbowl digits are "stationary". If I were to assume the winner's and loser's digits were independent (probably not true, but maybe close enough) I could get the joint likelihood for pairs with less data. I may do the analysis and see what it looks like, anyway...

Posted by: Marty | February 04, 2006 at 10:06 AM

BTW, FYI, the prizes are:

Quarterly winners - $50, $150, $250, $350 (for end of game)

Reverse digits - $50 each quarter and end of game.

8 prizes, $1000.

Posted by: Marty | February 04, 2006 at 10:07 AM

BTW Marty, I didn't mean to belittle your efforts. Very well done. Where else but Mazurland can you find such an in-depth analysis? I'll be rooting for you numbers as I watch the game. I'm also looking forward to Aretha Franklin singing the National Anthem.

I was surprised the number "1" didn't have a better showing. (21, 31)

Posted by: Chris | February 04, 2006 at 10:51 AM

It does now. The Steelers added a "1" to the winner's pile for end-of-game. I'll have a better analysis for next year's game. Time to start thinking about March Madness...

Posted by: Marty | February 07, 2006 at 09:10 AM