Nate Silver points out that just because the spread in today's Super Bowl is small (the Packers are something like a three-point favorite) doesn't mean that the game will necessarily be close. It just means that it's almost equally likely to be a blowout in one team's favor as in the other's.
Not surprisingly, though, the regression line for margin of victory, as predicted from point spread, is very close to having slope 1 and passing through the origin. As it should, because otherwise bettors would be able to take advantage of it! Say that 7-point favorites won, on average, by 9 points. Assume that the distribution of actual margin of victory, conditioned on point spread, is symmetrical; then half of 7-point favorites would win by 9 points or more, so more than half would win by 7 points or more, and one could make money by betting on them. On the other hand, say that 7-point favorites won, on average, by 5 points; then you could make money by betting agsinst them.
(For what it's worth, I don't have a particular interest in this game. In fact I probably won't even watch it. I have no connection to either Pittsburgh or Green Bay, and as longtime readers know, I'm a baseball fan.)
Showing posts with label gambling. Show all posts
Showing posts with label gambling. Show all posts
06 February 2011
23 July 2008
Rubik's cube hustling?
So I've finally memorized a solution to the Rubik's Cube. (I may be speaking too soon; let's see if the move I could never remember is still in my head tomorrow.)
I'm very slow, though. It's not the most efficient solution.
That got me thinking. There are pool hustlers, who act like they're no good at pool, start taking bets, and then all of a sudden are really good. If someone could solve the Rubik's cube really quickly, could they make money off it as a Rubik's cube hustler? Bring the cube somewhere where there are people, act like you can only solve it slowly, take bets, and then solve it quickly.
It just might work.
It would be crucial to find the right audience, though -- somewhere where people are familiar with the cube. So a bar, the typical place for pool hustling, wouldn't work. The right math department might work. But not mine -- I have readers within my department, and I'm pretty sure I've given too much away by making this post. Fortunately I don't have the skill to pull this off anyway.
I'm very slow, though. It's not the most efficient solution.
That got me thinking. There are pool hustlers, who act like they're no good at pool, start taking bets, and then all of a sudden are really good. If someone could solve the Rubik's cube really quickly, could they make money off it as a Rubik's cube hustler? Bring the cube somewhere where there are people, act like you can only solve it slowly, take bets, and then solve it quickly.
It just might work.
It would be crucial to find the right audience, though -- somewhere where people are familiar with the cube. So a bar, the typical place for pool hustling, wouldn't work. The right math department might work. But not mine -- I have readers within my department, and I'm pretty sure I've given too much away by making this post. Fortunately I don't have the skill to pull this off anyway.
09 July 2008
Lottery tickets with really bad odds
A CNN.com article talks about lottery tickets with zero probability of winning.
Why, you ask? Because some state lotteries continue selling the tickets for scratch-off games even after the top prize has been awarded. Therefore the odds stated on the ticket are, as of the time the ticket was purchased, incorrect.
But let's say that half the tickets for some game have already been sold, and the top prize not awarded -- then the tickets that are still out there have double the probability of winning that they did originally. You wouldn't see anybody complaining about that.
One way to fix this would be to have all the tickets be independent of each other, but drawn from the same distribution -- so instead of having one grand prize among the 100,000 tickets, each ticket independently has probability 0.00001 of being a grand prize ticket. But then there's a significant probability that there will be no grand prizes awarded, or that there would be two or more.
And some lottery websites actually state which prizes have already been awarded. So it might be possible for somebody to use this information to their advantage, by betting only in lotteries where a disproportionate number of prizes remain to be awarded. This is basically the same principle as card-counting in blackjack, where the player bets more when the cards in the deck are more favorable. I suspect, though, that this wouldn't work well because the house edge in lotteries is much higher than that in casinos.
Why, you ask? Because some state lotteries continue selling the tickets for scratch-off games even after the top prize has been awarded. Therefore the odds stated on the ticket are, as of the time the ticket was purchased, incorrect.
But let's say that half the tickets for some game have already been sold, and the top prize not awarded -- then the tickets that are still out there have double the probability of winning that they did originally. You wouldn't see anybody complaining about that.
One way to fix this would be to have all the tickets be independent of each other, but drawn from the same distribution -- so instead of having one grand prize among the 100,000 tickets, each ticket independently has probability 0.00001 of being a grand prize ticket. But then there's a significant probability that there will be no grand prizes awarded, or that there would be two or more.
And some lottery websites actually state which prizes have already been awarded. So it might be possible for somebody to use this information to their advantage, by betting only in lotteries where a disproportionate number of prizes remain to be awarded. This is basically the same principle as card-counting in blackjack, where the player bets more when the cards in the deck are more favorable. I suspect, though, that this wouldn't work well because the house edge in lotteries is much higher than that in casinos.
01 June 2008
Betting on his life
Man set to win on 'lifetime' bet, from the BBC.
In April 2007, a British man, Jon Matthews, was diagnosed with lung cancer and given nine months to live. He placed a bet with a bookmaker, William Hill, of 100 pounds at 50 to 1 odds. He's still alive, and he's collecting his 5000 pounds today.
Now, I have to think that the bookmaker offered odds which were much too long. Assuming that "nine months to live" actually means anything, I'd guess it's the median survival time. So this man had a probability of something like 1/2 of living the nine months. The bookmaker should have been offering 1 to 1 odds. (Actually, not even that, because they've got to make a profit!)
And even if it's the mean survival time that's meant, I doubt the distribution is sufficiently skewed to make a real difference. I suspect that "you've got X months to live" is just something that doctors say to patients informally, though, and it may not be sufficiently well-defined to quibble about this.
Of course, as the article points out, bets like this don't get made often. If they did, the bookies would pretty quickly figure out that they shouldn't be offering such large payoffs.
In April 2007, a British man, Jon Matthews, was diagnosed with lung cancer and given nine months to live. He placed a bet with a bookmaker, William Hill, of 100 pounds at 50 to 1 odds. He's still alive, and he's collecting his 5000 pounds today.
Now, I have to think that the bookmaker offered odds which were much too long. Assuming that "nine months to live" actually means anything, I'd guess it's the median survival time. So this man had a probability of something like 1/2 of living the nine months. The bookmaker should have been offering 1 to 1 odds. (Actually, not even that, because they've got to make a profit!)
And even if it's the mean survival time that's meant, I doubt the distribution is sufficiently skewed to make a real difference. I suspect that "you've got X months to live" is just something that doctors say to patients informally, though, and it may not be sufficiently well-defined to quibble about this.
Of course, as the article points out, bets like this don't get made often. If they did, the bookies would pretty quickly figure out that they shouldn't be offering such large payoffs.
19 May 2008
It's a tax on stupidity -- but nobody's that stupid.
You win $1,500 by feeding slot only $3.75 million -- Paul Carpenter, from the (Allentown, Pennsylvania) Morning Call, via fark.com.
From that headline, it sounds like slot machines essentially never pay out, right?
Carpenter becomes suspicious that the machines are rigged because Flo Fabrizio, a "key backer" of legalized gambling in Pennsylvania, won $1500 on his first play of the machine. Now, I can understand this suspicion -- sure, it could happen randomly. But then Carpenter goes on to do some "math" which proves he really doesn't understand what he's talking about. The article basically debunks itself.
It states that slot machines in Pennsylvania have to pay back at least 85 percent of what they take in. So then Carpenter says:
Could there be corruption in the gambling industry? Sure. But this doesn't prove anything. Not to mention that these calculations seem to imply that slot machines basically never pay out. Casinos aren't stupid -- they know that if slots essentially never paid out then nobody would play! The whole trick is that if you have enough small gains, you won't realize that on average you're losing.
On the other hand, they say that gambling is a tax on stupidity. At least this man's stupidity will keep him out of the casinos.
From that headline, it sounds like slot machines essentially never pay out, right?
Carpenter becomes suspicious that the machines are rigged because Flo Fabrizio, a "key backer" of legalized gambling in Pennsylvania, won $1500 on his first play of the machine. Now, I can understand this suspicion -- sure, it could happen randomly. But then Carpenter goes on to do some "math" which proves he really doesn't understand what he's talking about. The article basically debunks itself.
It states that slot machines in Pennsylvania have to pay back at least 85 percent of what they take in. So then Carpenter says:
If it takes a regular player 50 million three-second plays to get a big jackpot, it would take more than 11 years, playing 10 hours a day, every day. More important, if each play averages 50 cents, the player would have to put $25 million into the slot. The 85 percent payback would be $21,250,000. So, odds are, the cost of playing long enough to get a $1,500 jackpot would be $3,750,000.That's true. (The "50 million" comes from the regulations that say that casinos can set up their slots so that the jackpot only pays one time out of every 50 million.) But somehow I doubt a $1500 jackpot would be that rare. And more importantly, while the player is making those fifty million plays, they'll make a lot of small bets back. This is like saying that poker's a losing game because you only get a royal flush once every several hundred thousand hands. Or saying that you must not be broke because you didn't buy a million-dollar house you couldn't afford, but ignoring all the little expenses that add up. (Try telling your credit card company that!) You can't cherry-pick the tail of the distribution like that!
Could there be corruption in the gambling industry? Sure. But this doesn't prove anything. Not to mention that these calculations seem to imply that slot machines basically never pay out. Casinos aren't stupid -- they know that if slots essentially never paid out then nobody would play! The whole trick is that if you have enough small gains, you won't realize that on average you're losing.
On the other hand, they say that gambling is a tax on stupidity. At least this man's stupidity will keep him out of the casinos.
16 January 2008
Shuffling cards and total variation distance
David Speyer asks: what is total variation distance? Total variation distance is defined as follows: given two measures f and g on a space X, of total mass 1, the total variation distance is the maximum (strictly speaking, the supremum) of f(A) - g(A) over all subsets A of X. But this is only really enlightening if you're one of those people who buys the whole "probability theory is the study of measures of total mass 1" (I first heard this from Robin Pemantle but I don't know if it's original to him), which makes it sound like probability is strictly a subset of measure theory. It's not, because when you restrict to measures of total mass 1 a whole host of probabilistic intuition is valuable. Speyer gives an interpretation of the total variation distance in terms of gambling.
Speaking of gambling, you should shuffle a deck of cards seven times before using it; this comes from the famous paper of Bayer and Diaconis (Dave Bayer and Persi Diaconis, "Trailing the Dovetail Shuffle to its Lair". Ann. Appl. Probab. Volume 2, Number 2 (1992), 294-313), which Speyer was reading. (This was in the January 9, 1990 New York Times. Had this blog existed eighteen years ago, I surely would have mentioned this article. But there were no blogs back then, and I was not old enough to be reading mathematics journals.) I had incorrectly believed that this means that seven shuffles gave a uniform distribution over all possible arrangements of cards, but in fact the distribution isn't uniform; it's just "close enough", in the sense that the total variation distance is small.
I suspect I confused this with the result that "if a deck is perfectly shuffled eight times, the cards will be in the same order as they were before the shuffling"; thus you should stop at seven shuffles because on the eighth shuffle all your work will be for nought! But nobody shuffles perfectly.
Speaking of gambling, you should shuffle a deck of cards seven times before using it; this comes from the famous paper of Bayer and Diaconis (Dave Bayer and Persi Diaconis, "Trailing the Dovetail Shuffle to its Lair". Ann. Appl. Probab. Volume 2, Number 2 (1992), 294-313), which Speyer was reading. (This was in the January 9, 1990 New York Times. Had this blog existed eighteen years ago, I surely would have mentioned this article. But there were no blogs back then, and I was not old enough to be reading mathematics journals.) I had incorrectly believed that this means that seven shuffles gave a uniform distribution over all possible arrangements of cards, but in fact the distribution isn't uniform; it's just "close enough", in the sense that the total variation distance is small.
I suspect I confused this with the result that "if a deck is perfectly shuffled eight times, the cards will be in the same order as they were before the shuffling"; thus you should stop at seven shuffles because on the eighth shuffle all your work will be for nought! But nobody shuffles perfectly.
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