And making money in Vegas is a bad long term life plan. It doesn’t matter what your “strategy” is. These games of chance are built around a simple premise: no matter who you are or what you do, the more you play, the more you will tend to lose money.

Maybe the best way to drive home the ridiculousness of our relationship to the random is with comedy (thank you, Stephen Colbert, for the most mathematically sophisticated analysis I’ve seen on TV in a long time.)

Let’s review some facts about how random events work, which I’m trying to teach to 2nd-3rd graders at the moment, but which I’m pretty sure the majority of adults don’t know.

**Fact 1: A random event is not affected by what comes before it**.

This means that lottery numbers that have come up before are equally likely to come up again. The numbers that won last week’s lottery are as good a bet to play this week than any other set of numbers.* By the way, it also means that 1, 2, 3, 4, 5, 6 is as good a lottery ticket to pick (sorry, Colbert) as anything else*, so don’t disparage that choice.

Thus, ignore the advice I found on a randomly chosen lotto website: “Look at past winning numbers. If you really enjoy statistics and probability, look for patterns in these lottery tickets, and use that information for the best lottery numbers to pick.” Looking for predictive patterns in random noise is the ultimate fool’s errand.

*Of course, some numbers are better picks in that they are less likely to be picked by other people, so if you do win, you’re not too likely to have to share.

**Fact 2: Nothing you do affects the outcome of a random event.**

All your superstitions, knocking on wood, crossing your fingers, etc., don’t do anything. There’s a very strong human need to feel like we’re in control of our lives. Well, we’re in control of some things, but not as much as we’d like, and we have absolutely no control over which numbers come up on dice, cards, or spinning balls.

Now, there are ways to bias an event. I may be unable to predict whether it will be sunny or rainy on a given day next year (a random event, for all practical purposes), but my chances of a sunny day will be better in the summer. Random doesn’t necessarily mean even odds. Similarly, my students have experimented with spinning dice rather than rolling them to constrain the outcomes, and some claim that they can get their odds of a certain outcome up to 1/3 instead of 1/6.

**Fact 3: Most specific events in the future are fundamentally impossible to predict.**

Predicting the future in any precise way is impossible. Predicting weather in the future, predicting the stock market number precisely, predicting which horse will win a certain race, etc.–virtually impossible. Interestingly, if you change the question in the right way, it is possible to make much more accurate predictions. For example, I have no way of knowing whether a given coin toss will be heads or tails; however, if you flip 100,000 times, I’m confident roughly 50,000 of those flips (give or take) will be head. Similarly, I can’t predict the weather in the future, but I can probably get a reasonable read on the climate, and I have no idea what the stock market will do on a given day, but I may be able to give a pretty good assessment of where the economy as a whole is headed (though even though both of these questions are at the right scale, they’re still very difficult to answer).

The difficulty in predicting the future is a relatively new discovery, mathematically, but at this point, it’s been established in the field of chaos theory.

These are lessons I’d like the 2nd-3rd graders in my math circles to understand. I’d like them to let go of their superstitions, and try to understand how randomness really works. There is perhaps no lesson the next generation needs to learn more urgently. And here’s my prediction: those who understand probability and statistics will have huge advantages over those who don’t in the information economy, and those with a hard technical knowledge of both fields will always have job opportunities waiting for them.

I’m heading in a few minutes to meet with a group of kids that has been showing some real finesse for these ideas already. Lately we’ve been designing dice games to be tricky enough to appeal to others even though the games are biased in our favor. In other words, I’m teaching them to think like casinos. If I can succeed, maybe I can keep them from ever patronizing those places, too. Or buying a lottery ticket.

Because, face it, it’s just not a good bet.

## Comments 18

Author

The reading I did in researching this post was insanely depressing. For example:

great post!

There’s a bit of irony here in that you wrote “on a randomly chosen lotto website” but I suspect you did not in fact choose the website randomly.

(http://mathforum.org/kb/thread.jspa?forumID=67&threadID=1336343&messageID=4297991 has the scoop on my alter ego as the superhero Defender of the Random)

Author

That was intended as a joke. Maybe too deadpan?

Author

I like the phrase “literally at random,” because it misuses two commonly misused words in a very small space.

Awesome post! Thank you. Here’s something interesting…

The odds of winning the megamillions jackpot is about 1/175,711,536, but the payout was $656 million. That means each lottery ticket had an expected value of about $3.73. And they only cost $1! Well that makes it sound like you should have played. A LOT!

Here’s a related problem Richard Mann showed me…

It’s a proposition: You flip a coin, hoping for a streak of heads. The game ends when you flip tails. If you flip n heads in a row, then I’ll pay you $2^n. That means even if you flip a tails straight away you get $1. How much would you pay to play this game?

Is it worth $2? Would you pay $5? Would you pay $100?

Author

Very interesting questions, Paul. Richard’s really makes the question clear.

It’s a Pascal’s wager kind of situation: if you have a small chance of an infinitely good payout, should you pay any amount to take the risk?

At some point, I’d like to suggest that we have a breakdown in the theory, almost like Newtonian physics breaks down at large and small scales. The payout ceases to be meaningfully different once the numbers get too big (is being a trillionaire so different from being a quadrillionaire?). Of course, maybe that’s just a dodge.

Here’s the crazy thing about the wager above: It isn’t just a small chance of an infinitely good payout. The expected value is infinite!!!

It must come down to variance huh?

Author

Doesn’t a small chance of an infinite reward give an infinite expected value as well?

By the way, I just added your blog to our bloglist. I’ve really been enjoying it.

Very interesting. I would not want someone teaching my child to make her game ”appealing” to other kids. Sure, it will show them how casinos con people into playing but it doesn’t show them that it’s morally wrong to dup fellow mankind. Two wrongs don’t make a right. I don’t think it’s necessary to get them to think like casinos. However, showing them Facts 1, 2 and 3 in class and getting them to test it themselves with the dice is great! Good job!

Author

Evangeline, I’m very glad you brought this up–ethics in mathematics was one of the next things I was planning to blog about.

Rest assured that I think it’s absolutely critical to get students to use their mathematical powers only for good. However, I disagree about the utility of getting them to think like casinos, both from a mathematical and an ethical perspective. I think the play of putting themselves in the other’s shoes, even when the other is a less ethical actor, can be a powerful part of a lesson on many levels. In particular, the games become fun, and funny–the kids who put a challenge forward are met with a response of both laughter when the students break down the game, and also “your game is mean” response, a reinforcement of the group’s ethics. I think it’s this kind of flexible play that makes it possible to reinforce good ethical behavior, and I imagine that as kids start to see the casino games as mean games and the idea that someone would offer one as a joke, it becomes all the more unlikely that anyone would ever seriously do it.

But you have a point. I consider the ethical element of all this to be critical, but it would be easy to teach the wrong lessons. I’ll make sure I’m reinforcing this as much as I should be.

“The odds of winning the megamillions jackpot is about 1/175,711,536, but the payout was $656 million. That means each lottery ticket had an expected value of about $3.73.”

There have been times in the past when lotteries had a real positive expected value, such as the one described at http://www.lotterypost.com/news/93200 where someone actually did buy several copies of all 1000 combinations and earned 20%.

Here, though, I worry that with all the people buying tickets, you might have to split the jackpot. Also, there’s the fact that it’s paid out over a whole bunch of years, and the present value is only about half the claimed jackpot. Not to mention taxes. I think I read somewhere that if you took all the money now, then after taxes you’d “only” have about $100M left.

I would love to know how you presented the lesson about the dice games – my second grader LOVES making up his own games, and also loves math. I’d like to try this with him at home, but I’m not sure where to begin.

Thanks for the great blog.

It’s true that statistics cannot increase your actual odds of winning the lottery. They can however be used to mitigate your losses by choosing numbers that either have less chance of resulting in a shared jackpot or combinations of numbers that are statistically far less likely to be draw eg. all odds or all evens. As you can see from the Powerball Statistics page here > http://www.lottonumbers.net/powerball-odds-evens.asp a player choosing all odds or all evens would return a significantly lower amount of wins over an extended period than a player who regularly played 2 odds and 3 evens or 3 odds and 2 evens. Knowing this doesn’t help to win an actual jackpot but over a 10 year period would result in far more small wins resulting in smaller overall losses.

Author

Unless I’m misunderstanding something in what you’re saying, George, I don’t think it matters whether you choose evens or odds. It’s true that there are more combinations of evens and odds that win, but that’s only because there are more numbers of that form. If you make any individual choice, it has the same chance of winning. I suppose a player who regularly plays 2 odds and 3 evens would be more likely to be able to say “someone else who picked 2 odds and 3 evens won today” more often than their counterpart who played all odds. But that’s exactly the appearance of control and near-success that keeps people going back to the lottery.

Those who claim that a lottery is purely random – why are you so sure of that? “Scientists” say that? Well, the mediaeval scientists kept saying that the Sun was turning around the Earth…then, a Mr Kopernik said they were wrong!

Imagine in some 100 years, a scientist will prove that “random” doesn’t exist at all…

But, really, ladies and gentlemen, are you sure the life is random?

There’s a man that just got a prison sentence for scamming the lottery my question is if it is random how did he scam them and how did they send him to prison so since they did that why can’t I sue the lottery commission since they know the numbers already

Author

There are certain ways to scam the lottery, historically. Some of these are simple fraud, like changing the numbers on a ticket to make it look like a winner after the fact. Some are mathematical, like finding deep patterns in the serial numbers of scratch lotto tickets, though that’s not illegal. I’d need to know more about the details of the particular scam to know what the story was in this case.