Useful Math Tips for Gamblers
One of Robert A. Heinlein’s many aphorisms asserts the following.
“Anyone who cannot cope with mathematics is not fully human. At best, he is a tolerable subhuman who has learned to wear his shoes, bathe, and not make messes in the house.”
Bob — you were a great writer, but that is w-a-y harsh, dude.
First of all, I don’t wear shoes in the house. Also, I prefer a shower. And about that math thing — it’s not that I can’t cope with mathematics. It’s that I don’t want to, and you can’t make me.
You’re not the boss of me.
Okay. Now that I’ve gotten that out of my system, let’s move on. We’re all adults here, and if there’s one thing adults know and understand, it’s that you have to do an awful lot of things that you don’t like to be successful — even while performing tasks you might otherwise enjoy.
Many people, for instance, enjoy gambling — in far greater numbers than those who enjoy doing math.
And it’s probably this disparity that allows quack gambling schemes like the Martingale system to flourish while the gambler’s fallacy languishes in relative anonymity.
No, to truly function as a gambler and not simply as a member of the cargo cult of some bogus “system,” you’re going to need to do a little math.
Not much, I promise. But still — there will be math.
The Bet You Make Is Not Equal to the Bet You Take
The most important use of math (for the gambler, at least; a NASA astronaut might have other, more pressing uses for math) is in the initial choice of a game to play.
Once we’ve dispensed with the skill vs. chance differential (which has a math of its own but is too boring and complicated to get into here), we need to discover what the casino will pay on various bets.
Generally speaking, a casino will pay out approximately 90% of whatever it takes in — but that is a gross generalization, and anyone thinking it is true in all cases should not be permitted to operate heavy machinery.
Some games pay out close to 100% of what they take in, and there are some that actually give a tiny advantage (about 1%) to the gambler. This is true for some video poker games and is always dependent on the pay tables offered by the specific machines. The trick to success here is two-fold.
- You must find the machine that offers the proper pay tables
- You must play flawlessly
But video poker is at the high end of the “most favorable to the player” scale of gambling options.
Baccarat (yes, baccarat) offers a similarly slender advantage to the house (with wagers on the “banker” hand slightly more favored than bets on the “player” hand).
At the low end of the spectrum are the Big Wheel (for some reason, featured in every gambling movie ever) and keno, with house edges in the double-digit percentages for either.
So, at this point, our math skills need to be no more sophisticated than recognizing one number is larger than another and that smaller numbers (in terms of house advantage) are more desirable. Easy peasy.
Why Poker Is a Game of Skill
Next step up, but still one of the simplest examples of math being vitally important in gambling, also happens to be one of my favorite games: Texas hold’em.
What I like about the math involved with hold’em is it doesn’t take much work to understand both the value and the usefulness of understanding the odds.
First off, just in case you’re fresh off the boat from North Korea, Texas hold’em is a game played with two cards dealt face down to each player and then a field of five community cards that everyone at the table uses to make the best five-card poker hand possible.
The community cards are dealt thusly: three to start, after which some wagering takes place, then a fourth card is dealt (the “turn”), then a break for more wagering. Finally, the fifth community card is dealt (the “river”), after which there is a final round of betting.
Okay, now that that’s out of the way, let’s get on with why math is important in this game.
There are 169 possible hands in Texas hold’em. Each and every one of them can win the game, and each and every one of them can lose. Bet you’re feeling kind of sorry you made fun of Kenny Rogers now, aren’t you?
They say the worst hand (or pocket cards, in the vernacular) you can be dealt is 7-2 off-suit. It won’t support a straight, and it needs four other cards of one of its two suits to form — at best — a very weak 7-high flush that even my grandma could beat. Sure, the flop could include a pair of sevens, and the river might give you quads, but seriously, how often do you think that is going to happen? 1 in 595, that’s how often. Math is a harsh mistress.
The key to playing smart hold’em is understanding your chances of drawing the winning hand. If you play with a pocket of A-K off-suit, and the flop pairs your ace, you now have two outs for your hoped-for set of aces.
Incidentally, there’s a better than even chance one of your opponents has also been dealt an ace (at a nine-seat table, in fact, it’s not just “better than even”; it’s nearly 70% likely). It’s even possible one of your opponents has both of the remaining aces in his pocket, which leaves you “drawing dead” — in other words, in the game with no chance of winning.
Unsurprisingly, the odds that an opponent may have an ace in the pocket increases with the number of people playing. With two people playing (heads-up), the chance your opponent also has an ace is a mere 12%. With six people playing, it’s even money that if you were dealt an ace, one of your opponents was, too.
Now, I’m not trying to imply that the key to hold’em is guessing what your opponents’ pocket cards are.
It is, rather, the importance of determining whether there is a statistical likelihood of something good happening for you — in this case, the turn card is another ace.
What are the odds that this might happen?
Easy to figure out: There are four aces in the deck. You hold one of them in your pocket, and a second one just showed up in the three-card flop.
So how do you know whether your opponent has an ace (or even two) in his pocket cards? Simple: Just ask him. I mean, seriously, what possible reason would he have to lie?
Back to reality.
At this point, you know the value of five cards, leaving 47 mysteries.
Of those 47 mystery cards, two of them are aces. Drawing one of them at the turn is a 2:47 chance. Another way to look at that is out of 47 hands you play with an ace in your pocket, the turn will give you that third ace. The other 45 hands, it will not.
That doesn’t mean you won’t win the hand; your pair of aces at the flop might hold its value all the way to the river — but don’t go betting the farm on it. We could get into value bets and implied odds here, but we won’t, because I just want to alert you to the importance of math in gambling, not terrify you with advanced algorithms.
The odds in craps are easy to calculate. I mean, after all, you have two cubical dice with between one and six pips on their various sides. How hard can the math be?
The outcome of any roll is a number between two and twelve (assuming the shooter doesn’t toss the dice across the casino or eat one of them — which might be an interesting prop bet but has nothing to do with craps).
So, we have eleven legitimate outcomes for any roll of the dice.
Some of the numbers won’t come up as often as other numbers. There are more combinations that add up to seven than there are combinations that add up to four (and only one combination that adds up to either two or twelve).
An easy way to think of this is that the closer a desired number is to seven, the more likely it is to be rolled. A one or a twelve each has a less than 3% probability of occurring, while a six or an eight each has a nearly 14% probability of happening.
Incidentally, much of the confusion associated with craps has nothing to do with the math and much more to do with the bizarre number and names of many of the possible bets. Should you place a horn bet, or is a yo bet more your speed?
Here’s a tip: Until you pick up the lingo and learn all the various line and proposition bets available, bet the Don’t Pass bar. You’re betting the shooter won’t make his point before he rolls a seven. Sure, you won’t be popular among the other players, since you’re rooting against the shooter, but did you come here to find new friends or to win some money?
Kenny Rogers was right about another situation in gambling — knowing when to walk away and when to run.
Keno is a game of chance. There is no skill involved in playing it at all.
Oh, really? Well, would it interest you to know that video keno pays 142 for hitting 7 out of 10 numbers, but the odds of hitting 7 out of 10 in keno are 620 to 1?
Guess what math just did for us.
It showed us very clearly that we will have to bet 620 times to win 142. Six hundred and twenty bucks to win one hundred and forty-two. As they say on the internets: You do the math.
By the way, not to be cruel or anything, but video keno usually pays about 10,000 of whatever you’re betting for hitting 10 out of 10 numbers selected. If the machine paid out true odds, it would pay a little over 8.9 million.
Don’t look so shocked. Yes, it seems like a ridiculously paltry reward for beating ridiculously high odds against it happening. But we’re not taking into consideration all the other wins you had during those 8.9 million keno games. Not to worry — the casino has never forgotten them.
The truth is, if you do the math, the casino’s payout is roughly 65% to 80% overall, depending on the specific pay table (which varies widely between machines and venues). Sure, the casino has to take into account all the times it had to pay when you hit five out of the ten, six out of ten, seven out of — well, you get the idea.
Still, that house edge is not as galactically horrible as what the 8.9 million figure I cited earlier implies — but it’s no surprise birthday party from [insert your secret celebrity crush here], either.
Which way’s the exit from here, Kenny?
Pie Are Round, Cornbread Are Square
I’m always happy when I’m able to work a Texas Aggie joke into an article, however tenuous the link between it and what I’m writing about may be.
That notwithstanding, I hope I’ve been able to give you at least a glimpse at the importance of mathematics in gambling. I also hope I’ve shown you that, in this case, at least, math is not nearly as hard as your seventh-grade algebra teacher made it seem.