The Multi-State Lottery Association has recently changed the odds for PowerBall, so I thought I’d update my earlier post. Summary of earlier statements: even though PowerBall, like most lotteries, is rigged to return fifty cents on the dollar, there is some jackpot size at which point it is theoretically a better than 100% return on a bet. (Presuming the ability to play an infinite number of times; in all tests of this kind, a small number of trials will show anomalous results. In PowerBall, that basically means that everyone’s lifetime play will be statistically anomalous.)

The new rules have made it even harder to win the big jackpot, and likewise for the smaller wins. The sole concession to the player is that the five-ball hit without the PowerBall has been increased to $200,000, so now it’s theoretically possible to win a million dollars without hitting the PowerBall draw (five out of five with a 5x PowerPlay multiplier).

In any case, if you play PowerBall without making the PowerPlay bet ($2 for a ticket versus $1), you are getting incorrect odds and hence contributing to the state when the jackpot is below $232,292,812. (Assuming cash payout in one lump sum; variations on the value of this prize in relation to the advertised jackpot not accounted for. There is also a bonus pool on the 5 ball win with no PowerBall when the jackpot increases more than $25m in any one game; also not included.)

The PowerPlay still improves the overall bet—a $2 bet increases the return on any non-jackpot win at least 2x, up to 5x. There used to be 380% return on this bet for any win; this has been lowered to 350%. Regardless, the jackpot needs to be “only” $189,520,398 to be an even game with this bet.

What I find interesting about this is that it shows the utter irrationality of PowerBall players. That is, the rule of design of most gambling is to provide selective reinforcement; provide the player with small wins on the road to taking the house edge. This is difficult when the house edge is as huge as it is with state lotteries, but you can still work this into the design. With PowerBall, every revision of the game makes it harder to win, therefore creating much larger jackpots at the expense of the small wins that the players used to receive.

In other words, people play PowerBall when they can win a sizeable fraction of a billion dollars, and they stay away when the win is “only” say, $30 million. This is actually mathematically correct, but I doubt most players apply the mathematics to this. From the player perspective, is a win of $15 million any less life-changing than a win of $100 million? Yet, that’s what draws people in.

Likewise, they don’t seem to mind a string of losses like most gamblers. In the current game, if you play every game (104 times a year), if you made even a small win more than twice a year, you’d be lucky. Back in 1999 or so I won enough in four drawings straight to pick up another ticket and a pack of smokes. That should happen now once every 17,300 years or so.

So my guess is that the PowerBall attracts even more strongly a sort of non-gambler gambler, which is to say that the people buying tickets aren’t really buying tickets to win per se, but they’re buying a ticket to dream about what they’d do with the money for the days until the drawing. This has been measured before, and also indicates that twice a week is about as often as you want to run a game of this kind. It also suggests that in ten years we’ll be seeing lotteries with billion-to-one odds against and regular demiannual prizes of a few hundred million. Which should be enough for a few tanks of gas.

You know you’re playing too much poker (and are a bit too into zero-sum game theory) when you see a bumper sticker that says, “God bless the entire world, no exceptions”, and you think, “well, in that case, it’s a wash.”

A series of surreal experiences during a night gambling in Atlantic City.

First, a lengthy introduction

I have several upstanding friends who are rather dismissive of my favorite vice, which is to walk into a brightly lit casino and spend a long evening in an orgy of cigarettes, coffee, and gambling. One comment I remember particularly from someone who is generally a wise counsellor advised, “Gambling is a tax on the mathematically uneducated.”

Now, the true picture is a bit more complicated than that. You can get an edge over the other players at a poker table which exceeds the rake (the money that the casino takes out of each pot), and you can get a mathematical edge at blackjack through card-counting, both statistically provable. And there are craps games where the casino’s edge on you is insignificant (in the hundreths of a percent).

On the other hand, at most other casino games, you’re officially Just Another Schmuck. In roughly ascending order of stupidity, there’s nothing you can do to play “well” against a casino in pai gow poker, carribean stud poker, roulette, slots, sic bo, the big wheel, or keno. In all of these games you can play badly and make things worse, but you can’t play well and make things even. (And the worst games of all are state-run lotteries, which have a 50% edge for the state or greater; these are the same states that regulate private companies to pay better odds.)

An interesting side note on blackjack. With card counting, you can consistently get about a 1-2% edge on the “house”, which is our lingo for the casino. Perfect strategy without counting gives an edge of 2-4% to the house. The way most people play, though, the house has an edge of 30%.

Likewise with craps, perfect strategy gives the casino an edge of less than 1%, depending on the rules variations you’re playing. But the way most people play, the house edge is closer to 20%. This is why running a casino is a license to print money; casinos make more money than the odds would indicate, because people are too blinking stupid to play the games in their own favor.

Slot machines, 21st-century style

Now, I prefer playing craps and blackjack, but most of my gambling is in Atlantic City, where the minimum bet is $5. At those stakes, you can burn $200 at craps in less than 15 minutes, and at blackjack in less than an hour. This is always true even if you’re a good player; you can work out your odds over millions of dice throws or hands, but you can’t do diddly about the variance. This means that even though you know how you’ll do over the long run, you’ll never predict what happens in the short run; coin-flipping is an even game, but you can still lose 50 flips in a row.

So when I’m in A.C. with about a hundred bucks to play with, I usually play slots, even though the edge is 13%, far greater than the table games of craps and blackjack. (Slots at higher stakes are less tilted against you.) Now, if you’re not a gambler, what you’re picturing when I say slots is probably not what’s really there. Sure, there are still the three-reel machines with cherries, but the big thing in slots these days are video slots: video games that look like slot machines.

A key point about these slots is that, since they’re video games at heart, they almost always have what’s known as the “second-screen bonus”. That means that if you get some combination on the reels, the reels go away and you get some secondary game. Most slots are set up so that the bonus is really where the payback is.

The thing about table games is that if you don’t get taught by an expert, you’re probably not going to figure out optimum play strategy on your own. It took a Ph.D. mathematician, Edward Thorpe, to invent card counting. Over on the craps felt, optimum strategy requires you to ignore 95% of the bets on the table, but variance is going to make it nearly impossible for you to intuit this.

Slots, though, slots are simple. Put in your money, push the button, see what happens. There’s no such thing as being skilled at slots (with one notable exception, which I’ll write about tomorrow). It takes a special brand of idiot to be bad at slots.

Which sets the stage for the story.

Meet the idiots

So Mom and I are off on one of our gambling bonding experiences; after dropping $20 a piece on something called Wacky Fruits (don’t ask), we hit some video poker machines.

Mom tosses in a $10, hits a button to get her hand, but instead the machine makes a winning noise, and we see that her $10 has somehow become $17.50. This is before she’s played a hand, mind you.

The previous player had gotten up and walked away while the screen showed a winning hand. Press the button, boom, win 30 quarters. Someone had missed the “press the button” part.

Now, it’s not too uncommon to wander a casino and be able to take money that other people left behind. Here you’ll find a few bucks left in the tray; there you’ll see a dozen credits left in a machine. Once in a great while, on a slot machine which pays its bonus after N number of spins, you’ll see a machine which is guaranteed to pay you X dollars after you bet a fraction of X. (The two machines I know of which do this are Boom! and a variant of Sevens Wild that I’ve only seen in Vegas.) So Mom’s experience isn’t that rare.

But this is the first time I’ve seen it happen twice in one night.

On the way out an hour or so later, we pass by a machine where a woman is just sitting down. And lo and behold, it’s just sitting there, waiting to play its second-screen bonus. The previous player—nowhere in sight—hadn’t realized that they were about to win some money, got up, and walked away. This on a bonus screen that’s usually good for $20-30, and potentially can pay thousands of dollars.

I tell Mom to hang out for a minute. ‘Cause I know what’s about to happen, and I think it’s going to result in free money. When you’re in the middle of a spin, you can’t put money in a machine. This machine’s in the middle of a spin, a winning one. The woman doesn’t realize this, and she’s going to think it’s broken.

She tries to put in a twenty. No response. She tries a different twenty. Nothing. Helpful woman to her left leans over to assist, nothing happens. Both manage to miss the LCD screen just above eye level saying, “Play your bonus!”, the LED readout on the bottom left saying, “Hit the big key to play your bonus!”, and the big key itself which is dead smack center on front of the machine and glowing like a flashlight.

Woman gets up to another machine, breaks her twenty into quarters, comes back and tries to put quarters in the machine, which fall into the tray. (Middle of a game, can’t put in money.) Finally, faced with an obviously broken machine, she moves to a different slot. I walk up, and say (for the sake of the story I might have to tell security later), “Are you finished with this machine?” Response: “It’s broken, I can’t get it to work.”

So I sit down, put three quarters in for show (knowing they’ll just clunk to the bottom of the tray), hit the big blinking key, and win 144 quarters. A few more spins to show helpful woman, now to my left, that I “got it to work”, and walked away with a free bucket of quarters.

A word on my gambling ethos

I’ll hasten to add, if I see anyone put in their money and start to walk away from a bonus screen, I stop them and tell them, “you won, don’t leave!” I also give back craps bets to the dealers when they overpay me, and I’ve told blackjack dealers who tried to pay my 22 that they’ve miscounted. You don’t deliberately try to screw the other players, and I think it’s bad karma to cheat the house when the dealers make a mistake.

This is a different case; here’s someone who didn’t make the bet that resulted in the win, and apparently had the synaptic activity of a gnat. From the moment she sat down I had a good idea of the drama that was to unfold, and if her handbag had hit the gigantic flashlight button, she’d have found herself the happy recipient of some free money, as my mother had. But if you’re going to walk away from a machine you’ve just won on (player #1, whom I never saw), and you’re going to call a machine broken because it’s waiting to pay you, then I for one see nothing wrong with being the guy who gets paid.

The thing that never ceases to amaze me about casinos is how little the people there know about the games they’re playing. The casinos rig the games so they always win, even against players doing their best. (The exception to this is blackjack, but most casinos use anti-counting techniques to dilute the counter’s edge. Casinos themselves didn’t know counting was possible when they started offering the game.) But most players don’t know there’s a best way to play. They might as well just hand their wallets to the bellhop and go eat their free dinner.

I can understand that most people might not have the statistical background you need to learn why you never split 10s. But I’ve never seen a blackjack player get up and leave his chips on the table.

Tomorrow, the slot machine that breaks all the rules.