Lottery Scams: can you game these games of chance?

First up, a big thanks to Brett for inviting me to be part of this blogging project. When Brett first asked me I was like "Really? You and Jon are both Wall Street types. I'm just some schlub who does a podcast with a $5 microphone. What can I do?" Brett is under the impression I have a good eye for scams and suggested I write about such. Hey, I can do that. Intelligent people not only fall for woo and bad financial advice but they also fall for many, many get-rich-quick schemes. Think how many of those Nigerian bank scam emails you get a week or even a day. What's surprising is how many intelligent people fall for it. I mean, if you were getting two or three similar sounding emails a day or a week, why would you think "sure those other guys are scammers but this guy, Prince Stelco, is the real deal."

Anyway, my inaugural post isn't about Nigerian bank scammers but about the most common "something for (nearly) nothing" offer we encounter every day. The lottery.

I read about a study in Canada that reported some disturbingly large % of Canadians base their retirement plans on the assumption their going to win the lottery. Don't ask me to cite that report. I read about this news story about 8 YBG (Years Before Google). Since this is Canada, where nothing ever changes, I'm sure attitudes have not largely changed.

The most popular lottery in Canada is the 6/49. For the longest time I always thought 6 was the low end and 49 was the high end. In other words you had to pick numbers between 6 and 49. It was eventually pointed out to me by my so-called best friend (and not in the most polite manner) the 6 referred to the number of numbers you had to pick. You had to pick 6 numbers between 1 and 49. Okay that made more sense. My ignorance I can only blame on ignorance. I'm not a big lottery player. Or even a gambler. I sometimes enjoy a night of harness racing, which is one step above the dog track in terms of a classy ways to spend your Saturday night. But I mostly go to the harness racing track to eat french fries. Who knew longer you go without changing that fryalator oil, better (and darker) those fries get. Mmmmmm. Mmmmmm. I walk in the race track with $20 of betting money (and another $20 for racetrack grub) and I'm happy to walk out with $0 and an extra inch on my waist line to show for my night out.

Now I think I'm not much of a gambler because I have, if not an educated grasp of probability, an intuitive grasp. Actually, it's mostly the observation that harness racing tracks and lotteries don't stay in business decade after decade by shoveling money out the door. The house always wins. I see no evidence to disbelieve that maxim, barring instances of either cheating or better math knowledge.

Computer Consultant Beat the Montreal Casino

We hear a lot about MIT card counters breaking the bank but there's a less well known story that took place in Canada. It's the story of Daniel Corriveau. Back in 1994 a Canadian computer consultant named Daniel Corriveau noticed the newly opened casino in Montreal (which is in Quebec, the troublesome Canadian province) had a computer operated Keno game. Montreal's casino wasn't a 24 hour operation and shut down at night and opened anew in the morning. Corriveau reasoned the computer running the game would be turned on roughly at the same time every morning. He also observed some reoccurring patterns in the Keno results and suspected the computer was using the same seed each time it was turned on. Using a crappy 286 computer at home, he was able to figure out which numbers were likely to come up and when. He and some family members got some cash together and then they hit three Keno jackpots and took home $600,000. The casino didn't let them walk out the door with the loot. They well understood this string of luck was only slightly more likely than all the subatomic particles in your car suddenly making a quantum leap from the parking permissible side of the street to the no parking side of the street. (There's a legend that this argument was made in a Canadian court by a physics student trying to get out of a ticket, but the judge wasn't buying.)

Anyway, the casino demanded Corriveau explain his amazing luck. Corriveau made up a story about using advances in chaos theory to beat the house. Initial news reports repeated Corriveau's claim without actually checking with anyone who knew about chaos theory (I guess it was just a great buzz word at the time). If an obscure computer consultant could implement some obscure branch of chaos theory to beat a Keno machine, it would have had profound implications on the security of passwords, the stock market, banking, cryptography, you name it. Even today the story still somewhat lives that Corriveau used some hyper dimensional math like block transfer computations to win at Keno. I suspect Corriveau didn't want to tell the casino people how he actually won, as Corriveau probably wanted to go back the next day and win again. I would. The casino eventually got the real story and went through the gambling regulations with a fine tooth comb. They looked at it upside down, under a full moon, and with a team of lawyers and soothsayers, but eventually they had to conclude that Corriveau broke no actual regulations and awarded him and his family the cash. I'm given to understand Corriveau was something of a folk hero in Quebec and a great deal of public pressure came to bear on the casino to pay Corriveau his winnings.

Most of us won't be so lucky to be both a computer consultant and frequent a newly opened casino run by idiots. Our only hope is the lottery. A few weeks back (Aug 6, 2008 to be exact) Canada was in the grips of lottery fever. The minimum payout for 6/49 is about $3 million but like some lotteries in the USA, if no one wins the jackpot, the winnings are rolled over into the next draw. As it happened, the Aug 6 jackpot was about $44 million. I should point out lotteries in Canada are treated in spirit (although not legally) like a "voluntary tax" (or more likely "a tax on stupidity"). The first attempt to introduce a legal lottery was in Montreal in 1968. It was spun as a voluntary tax by the crafty mayor. Since no government (with maybe the exceptions of Sweden) has the chutzpah to tax a tax, lottery winnings are tax free. They are also paid out as a lump sum in the full amount. They're not paid out over 20 years with a lower lump sum option. So in the USA a $44 million jackpot, after you take the lower lump sum payment and shell out for taxes, you're probably looking at $18 million or something. Better than a kick in the balls, I suppose. But right. In Canada it's $44 million and Bob's your uncle. Unless of course there are multiple winners. Then the prize is split. But we'll get to that point later, much later. And the US and Canadian dollar are nearly at par these days and Canadian taxes are a lot lower now and we still have the free healthcare. The point I'm driving at here is $44 million is pretty much real money, even if it is Canadian and looks more like large postage stamps than the American ideal of real money.

Can you game the lottery?

When jackpots get big, people get excited, more people play, and of course people decide there must be a way to game the system, like our friend Corriveau gamed the casino. For most people, their brilliant plan is to simply buy a whole lot of tickets. To the 6/49 lottery people's credit, they've made it a mantra in 6/49 ads that the chances of winning are 1 in 14 million (actually it's more like 1 in 13.9 million but let's call it 1 in 14 million).

Like I say, most people decide the best way to game the lottery is buy a lot of tickets. More tickets. More chances to win. (If 50 mg of Vitamin C is good then 10,000 mg of Vitamin C must be better…) A friend was planning on dropping $100 on the Aug 6 draw. When friends announce these kinds of plans I like to point out the following odds. If you bought two tickets a week (there are two draws per week), that's 104 tickets a year. I'll remind you your chances are 1 in 14 million. So you have a 104/14 million chance of winning per year. Let's reduce that fraction. You have a 1 in 134,615 chance of winning that year (in percentage that's a .000074% chance).

You need to keep playing for about 134,615 years before you're likely to hit the jackpot. Even Queen Elizabeth isn't going to live that long. Even if I did live that long, I'm not so certain I'm going to be in the mood to leave my house, chip through a couple ice ages of advancing glaciers, to visit the lotto shop. If you started dropping $100 a ticket (tickets are $2 a shot so you're jacking up the odds of winning 50 x), you're going to win after about 2700 years of playing. Again, not in your life time.

If you ever want to test how long it will take you to win the lottery check out the lottorobics simulation.

Now reasonably one can't win if you don't play. And reasonably if you spend $2 a draw or $4 a week that's about what you'd spend on a latte at Starbucks. It's not huge money. But if you're dropping $200 a week on the lottery, that's rent. That's a retirement nest egg. If you're dropping $100 a draw that's $200 a week or $10,400 a year. If you put that $10,400 in a low risk investment and kept adding $800 a month, at 2% annual interest you'd have $312,375.32 after 25 years.

Surely My Hyperlobic Omni-Cognate Neutron Wrangler Can Help?

Other people think they can game the lottery by using computers to increase their chances of winning. There are lots of programs on the Internet you can download that promise gee whiz sounding statistics ("wheeling systems") can increase your odds. Some are freeware. Some are shareware. The shareware programs are a waste of money and the freeware programs I'd bet are full of adware and Trojans. I would imagine in some multi-level hell for very stupid people on the internet, where the lowest level of hell is full of people who would actually click on an EXE file thinking they'll see Britney Spears in a XXX hidden video, people who would download lottery software are about third or fourth from the bottom. They're not likely to notice you've slipped a Trojan onto their system and are now blasting out EXE files to other stupid people offering the aforementioned Spears XXX action.

One such lottery program I looked at was quite the hoot in terms of claims and statistical woo. Actually it wasn't so much woo that caught my attention as how the advertiser simply stated the obvious but made it seem like it was a big revelation. For example, the program claimed it could pick winning numbers. This was guaranteed even. Hmmm. A daring claim? Well, I guarantee you 7,12,25,28, 35, 48 or even 1, 2, 3, 4, 5, 6 is going to win. Eventually. I would be remiss if I didn't point out 1, 2, 3, 4, 5, 6 is as likely a winner as 7, 12, 25, 28, 35, 48. Every number has an equal probability of being include. What are the odds the number 1 is going to be drawn? 1 in 49. What are the odds 7 is going to be drawn? 1 in 49. If 1 is drawn, does the next number care at all what came before it? Of course not. 2 or 12 now have equal chances of being drawn.

This lottery program also claimed it could increase your chances of winning. And it made another guarantee. Its method? It helped you pick more than one number. If it helps you pick two numbers, why, you've doubled your chances. Money well spent, no?

Another claim made by this program, revealed in a kind of "secrets the lottery corporation doesn't want you to know about" fashion, is that over 50% of winning numbers have at least 2 consecutive numbers. So over 50% of winning tickets will have, say, a 12 followed by a 13 or 45 followed by 46. There's a greater than 50% chance that two (out of 6) will be consecutive. The ad wizard writing the copy claimed this secret was teased out after the program creator ran complex statistical analyses over the course of a whole year. Golly. Took him a whole year to figure that out? We can figure it out in about 2 minutes.

Suppose the first number drawn is 12. We need a 11 or 13 as the second number drawn to create a two consecutive number result. The odds that the second number is going to be 11 or 13 is roughly 4% (2/48 and not 2/49 as there can be no repeating numbers, so there is now one less possible number). Let's say the second number drawn is 4. That's not 11 or 13 and that doesn't much help us. But no worries, mate. We now pin our hopes on the third number, which can be 3, 5, 11, or 13. That's roughly a 8.5% (4/47) chance. Suppose our third number comes up 40. If the fourth number is 3, 5, 11, 13, 39, or 41 it will satisfy our two consecutive number condition. That's a 13% (6/46) chance. Our fifth number has 8 out of 45 numbers to satisfy the condition (about 18%). Our 6th and final number has 10 out of 44 possible numbers that will create a two consecutive number combination. That's roughly a 23% chance. If I remember Stats 201 correctly, all those percentages get added to calculate the chance any position will satisfy the condition. So, pure chance assures roughly 66% of winning numbers will, over time, have two consecutive numbers. I'm not really taking into account that 2% of the results will have a 49. If 49 is drawn then only 48 satisfies the condition. But I don't think this is going to radically alter the 66% figure.

Let's Check

Right. We can even check this at the Lottery Buddy site. Looking at a list of winners for 1994 and 1995, looking at the first 40 draws, 21 (52.5%) had 2 consecutive numbers. That's not quite 66% but real life results regress toward the theoretical results as you increase the sample size. Consider a 50/50 coin flip. We're not assured two heads and two tails after 4 flips. After 4,000 flips we still might not be at 2,000 heads and 2,000 tails. We'll be closer to the ideal but doubtful we'll be spot on.

You can test this here. Generate however many flips you want. In the Integer fields enter 1 and 2 (1 being a Head and 2 being a Tail). For 4,000 flips I got 1937 Heads (expected 2,000). For 400 flips I got 220 Heads (expected 200). For 4 flips I got 1 Head (expected 2). I'll leave it up to you to guess how I quickly discovered I had 1937 one's in a page of 4,000 numbers. Hint: My favorite application is Microsoft Word.

Anyway, coming back to the 2 consecutive number claim. Knowing 66% of winning numbers should have two consecutive numbers, the question is: This helps us how exactly? It doesn't. At all. You need to ask yourself which two numbers need to be consecutive. Remarkably there are 48 possible number pairs to choose from. Got any insights?

Alright, well people might approach it this way: If your favorite lottery number lacks a consecutive number pair does this mean you have a lower chance of winning? Theoretically 66% of winners should have two consecutive numbers. The answer is, of course, no. Look at it this way, most people are not born on a weekend (5/7 vs. 2/7). Are your odds better, then, your child is going to be born on a Monday and not Saturday?

Another trick some people employ is finding on the Internet tables of which lottery numbers have won in the past. The Ontario Lottery Corporation fearlessly (that should be a clue) publishes number performance here.

As of this writing, the number 31 has been drawn 376 times and that laggard 15 has been drawn only 297 times. Since each number has a 1/49 chance of being picked, probability dictates each number should have been picked an average of 332.6 times (16296 draws / 49). The number 31 has been picked 43 times more often and 15, clearly the red-headed stepchild of lotteries everywhere, has been picked 35 times less than probability dictates.

Now here is where the "gambler fallacy" can go one of two ways, depending on your faulty beliefs about how the world is supposed to work. Should you play the numbers that have been draw the most, because these numbers are "hot"? Or should you play the laggards like 15 because they're long over due?

Or does this make a difference? As I noted above, as you increase the number of coin flip trials the observed results should approach the expected results. If you have 12 trials and get 11 heads and 1 tail, that seems pretty suspicious. But what if you get 8 heads and 4 tails? If you did 12,000 trials and got 11,000 heads and 1,000 tails, you could quite reasonably conclude something was up. But what if you had 8,000 heads and 4,000 tails? What if you had 800,000 heads and 400,000 tails? As noted above, as we increase trials we should be approaching the expected result. So 8H vs 4T isn't that suspicious. The results from 12 trials shouldn't approach the expected results very much. 8,000H vs 4,000T is looking much odder as 12,000 trials should have approached the expected results a lot more. 800,000H vs 400,000T is looking very very odd. 1.2 million trials should approach the expected results a lot closer than 12 or 12,000 trials.

So how do we know if some distribution is likely chance or likely the lottery people playing around with balls of different weight? Some second year university stats called the Chi Square test comes to the rescue. The Chi Square test helps us compare expected outcomes (6,000 H vs. 6,000 T) with observed (8,000 H vs 4,000 T) and determines if the observed outcome deviates too far from the expected outcome given the trial size.

Unfortunately, last time I did a Chi Square test was in 1986. I'm pretty old these days and a little rusty (by a little rusty I mean rusty like a Ford V8 engine plant in Michigan). But no problem, because we have both the Internet and Excel, which we didn't have in 1986. At least not at my crap university (Sir Arthur Meighan University).

Alright what we do is we create two columns in Excel, the observed results and the expected results. The observed results are from the lottery page. We plunk all 49 results into one column. The expected results are the same for each cell in the column, 332.6 (16296 draws / 49). Then we use the CHITEST() function in Excel. And Excel reports back .395249. Hmmm. Yes. Very impressive. A .395249. Hrm. What's that mean? Well Googling we find that's the p-value. You might recall when skeptics are talking about ESP results they throw around this p- value term a lot. That's basically the chances the results can be explained by random chance versus an effect. Many people in science use a p-value of .05. If the calculated p-value is less than .05 then you can reasonably conclude the results are not a product of chance. If it's more than .05 then you can reasonably conclude the results are chance. So .395249 from our CHITEST() is quite a bit bigger than .05. So we can quite safely conclude the lottery number distribution falls firmly under a chance distribution.

[caption id="attachment_237" align="alignnone" width="264" caption="Dumping those lottery numbers into Excel"][/caption]

If you don't have Excel you can do it at this page. You can work through the basics of Chi Square here.

Wheeling system

The final bit of lottery software woo is the "wheeling system". The idea here is instead of playing, say, 6 different number combinations (or lines), you select 7 distinct numbers (say 1, 2, 3, 4, 5, 6, 7) and play permutations:

Example 1:

1 2 3 4 5 6
1 2 3 4 5 7
1 2 3 4 6 7
1 2 3 5 6 7
1 2 4 5 6 7
1 3 4 5 6 7

You have a 6/14 million chance of winning the jackpot. But, the logic goes, what if you score 3 or 4 of the numbers?

Well, let's answer that. So what if the actual draw is 1 2 3 4 21 46?

You check your wheel and find you've hit 3 and 4 "ball" winners thusly:

Example 2:

1 2 3 4 5 6
1 2 3 4 5 7
1 2 3 4 6 7
1 2 3 5 6 7
1 2 4 5 6 7
1 3 4 5 6 7

So you hit three 4-ball winners and one 3-ball winner. According to the latest 6/49 results a 4-ball winner pays about $75 and a 3-ball pays $10. So that’s 3 x $75 + 1 x $10 for a take home of $235.

So what's the problem here? First your odds of a single line (any line) winning 4/6 numbers is 1 in 1033 or about .01%. Your odds of a single line winning 3/6 numbers is 1 in 57 or about 1.75%. Let's say you bought 6 tickets, all with the exact same number.

Example 3:

1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6

Pretty boring, huh?

You're certainly not increasing your odds of winning the jackpot. If you hit the jackpot (and assuming you're the only winner), you won't get a jackpot six times greater. You'll only win the jackpot.

So let's say you hit a 4-ball win. The numbers come up: 1 2 3 4 12 42.

I'm sure I don't have to illustrate it but we will. We win thusly:

Example 4:

1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6

So we get 6 x $75 or $450.

But let's say you decide you want to double your chances of winning a 4 ball (which is 1 in 1033) and you can live with a pot half the size, $225.

So you play thusly:

Example 5:

1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 7 8 9
1 2 3 7 8 9
1 2 3 7 8 9

You've now doubled the chances of winning a 4 ball. Your odds are now 1 in 516.5. Of course you've also halved your win.

Okay but let's say you want to double the chances again and you can again live with half as much in winnings ($112). So you play:

Example 6:

1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 7 8 9
1 2 3 7 8 9
7 8 9 10 11 12
7 8 9 10 11 12

So what I believe you're ultimately doing with a wheeling system is, in a sense, creating new prize categories. The lottery pays out for a 4 ball win with odds of 1 in 1033. The lottery pays out for a 3 ball win with odds of 1 in 57. Notice there's nothing in between? With a wheeling system you're kind of inserting prize categories. You're trading off a higher cash prize (as in Example 3) for odds better than 1 in 1033 but with a lower prize.

But I spend all my money on new iPods, how can I ensure my retirement via the lottery?

You have only really two levers of control in the lottery. As noted above, if you play more numbers you increase your odds of winning. But as noted above, you can lay out substantial amounts of cash and still not increase your odds in any meaningful way.

The only other thing left to you is doing what you can to ensure if you do win the jackpot you don't share the jackpot. It's not uncommon to have multiple jackpot winners. Wouldn't it be a bitch if you had to share a $44 million prize with two others who picked your exact same numbers? Okay well technically being given a check for $14.7 million isn't a bitch but still. A lower prize like $3 million could mean taking retirement at 40 with $3 million in the bank or retirement at 55 with only a $1 million in the bank.

So how can you reduce the chances of not sharing a jackpot? First understand human psychology. Some people will probably play 1 2 3 4 5 6 for shoe shines 'n' giggles. I bet there's a load of people who play Hurley's numbers from Lost. Maybe some people hope their master Satan will reward them for faithfully playing 6, 16, 18, 26, 36, 46 (all the numbers with 6 and 18 because 666 adds up to 18). As noted above, some people will try to play "hot" numbers and some people will try to play "cold" numbers thinking they're due.

Other people might believe a number like 1 2 3 4 5 6 could never come up and want their number to look random. Most people would say a series of coin flips that produces "TTTHHH" is less likely to be random than "TTHTHH" which just appears random. Of course both are as likely. So people in their efforts to hand code a random sequence might largely duplicate each other's efforts.

End of the day, the best way to reduce the odds you'll choose the same number as another guy is simply let the lottery machine's "quick pick" function pick your number.

Oh, and that 1, 2, 3, 4, 5, 6 number combination? At the Lottorobics page I ran over 1,000 draw on 1, 2, 3, 4, 5, 6 and a randomly generated series (8, 12, 13, 26, 39, 48). Guess what? After spending $2082 on tickets, my highly improbable combination produced about $230 in winnings. Okay, I let it run on the randomly generated combination. Guess what? After 1042 draws, I pocketed $220. I actually did better with 1, 2, 3, 4, 5, 6!

-- Karl Mamer