*I’m in the wild blue yonder today; so here is a distraction. Thanks to reader Jade for suggesting the topic.*

Nobody can scratch better than Las Vegas resident Joan Ginther, who has just scrapped that little gray fuzz off of her fourth winning lottery ticket. Her *fourth* win!

The two questions on everybody’s mind are: How do I hit Ginther up for a loan? and How do I work her magic? I don’t have a sure answer for the first, other than to say that, she being female, flattery rarely fails; but I can tell you all about the chances of duplicating her performance.

First, her achievements, according to the (as it is sometimes miscalled) Corpus Crispi Caller and Yahoo Buzz:

- 1993: $5.4 million (paid in yearly installments). Odds: 1 in 15.8 million,
- 2006: $2 million (lump-sum payoff). Odds: 1 in 1,028,338,
- 2008: $3 million (lump-sum payoff). Odds: 1 in 909,000,
- 2010: $10 million (lump-sum payoff). Odds: 1 in 1,200,000.

It’s not clear if these are the pre-government confiscation amounts, or the actual dollars she pocketed; probably the former. Still, even considering the (approximate) 40% federal tax bracket, if the lovely Ginther has been living clean, then she likely has at least has several million in the bank.

But since she’s been camped out in Vegas, and she has quite positively evinced a love of gambling, she might not have much left after all. For to win that many lotteries requires her to buy many, many tickets.

Let’s simplify a bit, just to make it easier on ourselves. The probability of winning her lotteries are approximately 1 in 15 million, and three 1 in a millions. I’ll assume the 1 in 15 million was a “bouncing ball” lottery, and that the others are all scratch-off tickets: the kind of gamble doesn’t matter in calculating the odds of winning, but naming it makes it easier to describe. We don’t know, but it’s a good guess that she likely bought more than one ticket per game.

Take her 2006 win. If she bought just one ticket for that gamble, then she had a 1 in a million chance of winning. If she bought two tickets, then she roughly doubles her chance of winning. If she was like a lot of folks I see lining up at the bodega windows, she might have laid down as much as a 100 bets in a few months’ time. Buying that many tickets pushes the chance of winning to 1 in a 100 thousand, a substantial jump.

There are about 13 years (we don’t know the exact dates of her wins) between her jackpot payout of 1993 and her next winning ticket in 2006. Assuming she bought 100 tickets a year—a not uncommon figurel; probably on the low side—then she might have racked up 1300 tickets. That gave her a just over a 1 in a 1000 chance of winning. Pretty good odds! If she bought 200 tickets a year, her odds of winning rise to almost 3 in a 1000.

Anyway, she did win in 2006, then she won again in 2008, which, of course, is only two years later. How many tickets did our Joan buy in those two years? We can only guess: but she had a pocketful of money, so, at least as a first approximation, we can imagine she bought another 1300 tickets. That gave the odds of winning (in 2008) 1 in a 1000 again.

And the same thing, or something like it, is true for her last win in 2010. That is, she likely had a 1 in 1000 chance of winning the last payout.

The lottery has no memory, by which I mean that winning before does not affect the probability of winning again. Given that and the rule that chances multiply, we can calculate that Ginther had a 1 in a billion (which is 1 in 1000 multiplied by itself three times) chance of winning her last three payouts. If she bought twice as many tickets as we guessed, then she had about a 2 in 100 million chance of winning thrice.

But what about her first win? It’s the same process. We have to make a guess about how many tickets she bought. It’s not impossible to imagine, this being her first win, that she dropped a substantial bundle before seeing her numbers come up. Say she blew six grand: that gave her the odds of 4 in 10,000 of taking home the jackpot.

Altogether, this makes the chances anywhere from 7 in a trillion to 9 in 10 trillion of winning four times, depending on the number of tickets purchased. Even if we assume she bought twice as many tickets as we guessed, this still works out to about 1 in 10 billion.

But that’s just the odds that she, Joan Ginther, wins four times. The odds that *somebody* wins that many times is much, much higher. As much as 2 in a 1000, if there were 20 million inveterate gamblers like Joan out there. And if there were 100 million—a distinct possibility: remember, we’re talking about many decades of lotteries from which to find four winners—then the chance of at least one Joan Ginther is about 1 in a 100.

Which suddenly doesn’t seem so small.

7 July 2010 at 6:27 am

Exactly how do winning odds accumulate across drawings (or in the case of scratch cards, printing lots)?

7 July 2010 at 7:40 am

Are you sure about this? I’m not a statistician to the stars, but your numbers seem off.

If the first win is a lottery rather than a scratch ticket game, then there’s a (let’s assume) weekly drawing. She’d have to buy her 6K of tickets for a single drawing to lower her chances that much.

As for the scratch tickets, these are preprinted with (say) one major winner in a fixed number of tickets. For her first scratch card win, she’d have to buy 100 tickets per year for the same scratch card set to lower her chances to 1:1000. It’s highly unlikely there’d be one set going for 13 years.

However, scratch card games do have a memory. For every loser, there’s a better chance to win the major prize, until it’s won and the chance for remaining cards goes to 0. So paying attention to how long a game has been going without having the major winner is enough to substantially modify chances.

7 July 2010 at 7:49 am

Its said that lottery winners — those that win really big payoffs ($1M+) don’t do so well financially in the long run, as a general rule. Especially when they take the lump sum payout. Apparently most winners are financial rubes that blow the loot on toys & so forth…

I’d be interested in a credible study on the point. Some [many?] years ago a fiction novel (I forget the title) came out in which a reporter was researching the above trend’s validity and in the story this was borne out with 10 very notable exceptions involving people that were particularly inept 7 who ought not have done so well financially over time. Turned out some genius figured out how to tamper with the balls to force certain winning number to come up, and get away with it via a secret deal with the clods he’d picked to buy the winning tickets. Things were just fine until the reporter came along asking questions & digging around…at which point the story degenerated into the usual deadly cat & mouse sort of farfetched chase of the sort that such novels require.

For people who like such stories, it was a good story …. if anybody recalls the title…

7 July 2010 at 8:18 am

I also wonder if the scratch ticket odds work as you’ve described. If there’s a game with 1M tickets, someone buys 100 tickets, and finds them to be all losers, what are the odds for winning in the next set of 100 tickets bought? Is it 200:1000000 (all tickets still have equal chance) or is it 100:999900 (remove the known losers, take 100 chances of the remaining set)?

7 July 2010 at 8:27 am

Ken,

Happened for real in Pittsburgh back in the 70′s IIRC. A local TV station had the contract to run the daily numbers game. Seems the announcer was in on a scheme to rig it. He eventually was caught and went to prison.

mt,

Interesting but you would need to know when a series started. Around here, the tickets aren’t simply purchased as stock but are provided by the lottery commission (via contractor). There’s nothing preventing them from changing the ticket set periodically. Needless to say that when they do it’s not publicized.

7 July 2010 at 8:34 am

The answer is B: remove the known losers, take 100 chances of the remaining set.

7 July 2010 at 8:37 am

She’s hacking.

7 July 2010 at 8:43 am

Interesting to note here that the winnings exceed the inverse of the odds. Which is to say that if these tickets cost $1, the expected value of the ticket was way higher than the cost of the ticket (more than 8x higher for the fourth win). With calculations like that, buying lotto tickets may well be more an investment than a gamble. However, it also means it’s the kind of game on which lottos lose money. The kind of game which cannot exist. So, either the numbers are wrong, the costs of the scratch-offs was high, or I REALLY need to play the Vegas lotto.

7 July 2010 at 10:55 am

So you’re saying that you need to buy 100 tickets or more to make a difference in your chances of winning the lottery? If you’re right, it suggests that when I buy a second ticket, I’m simply throwing my money away! What awful news!

7 July 2010 at 11:52 am

My guess is she’s not playing one-dollar scratchers.

My wife loves it when I bring her home some of those as a fun surprise every once in a blue moon, and if Vegas is anything like Orange County, one-dollar scratchers are hard to come by as they’ve been all but replaced by predominately three-dollar and two-dollar scratchers.

Probably needless to say, but the blue moons have been a bit further apart out here.

7 July 2010 at 12:19 pm

With a naive belief in one’s own luck (reinforced by winning a $5.4 million jackpot) and a ton of money, to the lucky person it would not seem unreasonable to invest several hundred thousand dollars in lottery tickets. Forget about 100 or 200 tickets per year.

It would be interesting to know what the best “investment” strategy would be if you were going to blow out really serious money on lotteries. In fact, Matt, this could be a new consulting area for you! Statistician to the lucky lottery winners! Lottery millionaires are waiting to contact you!

Years ago I had the fun and profitable experience of doing some Monte Carlo lottery game simulations on a supercomputer for one of the companies that prints the lottery tickets. They wanted to build a statistical database that would allow them to predict how many pink Cadillacs and trips to Hawaii would be given away by a given batch of game cards. If you know enough about how the games work and how the card batches are played out, you may be able to shift the odds a little.

7 July 2010 at 7:09 pm

Your odds make an assumption that the person redeeming multiple winning lottery ticket also bought the tickets. This is not an assumption the IRS and Interpol always make. Lottery tickets have been used as a vehicle to launder money.