Proof that you must play the right poker limits

• Mike Caro   Exit

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Note: Not at the old Poker1 site. A version of this entry was originally published (1999) in The Gambler magazine. Updated 2023 to further clarify the summary.

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Today’s lesson is important to all serious gamblers. Today we’re indirectly going to prove a poker point by scarcely talking about poker.

You enter a casino. You’ve chosen roulette or maybe craps as your favorite game. You’re ready to fight the odds.

Fine. You’ve come armed with $1,000 disposable money and you’re prepared mentally to lose it. If you win, that’s all the better. Your main mission today is to have a good time and to hope for a win.

But, because you follow my advice, you know that there is no magic formula you can apply to the amounts of your bets that will overcome the house edge. Can’t be done. You also know that picking red or black or any combination of numbers at roulette won’t help you. Those odds are in the casino’s favor and they will stay that way.

You understand the risks

Still, in the short term, you have a chance to win. You just need to get lucky. In your mind the casino is entitled to expect a long-range profit as long as they provide a fair game and a reasonable chance for you to win today while being entertained. You aren’t deceiving yourself. You know you aren’t going to be able to win in the long run and play craps or roulette professionally. But you’re willing to play recreationally with your $1,000 anyway. (Actually, I personally don’t treat gambling this way, but I see nothing wrong with players who want to test casino games for enjoyment. Me, I usually stick to games that give me an advantage.)

You know that if you want to win in the long run, you’ll have to master a game where the odds aren’t fixed against you, such as poker. In poker it’s player against player, instead of you against the house, so you have a chance to win over the long run. But it takes a lot of skill, and that’s not the challenge you’re looking for today.

So, with your eyes wide open, realizing you need some luck, you decide to bump heads with the casino at craps or roulette. And this brings us to today’s great question in our quest to understand the very nature of gambling from a logical point of view: Are you better off making big bets or small bets?

How much should you bet?

Let’s say you predetermine that you’re going to gamble at roulette until you either double your money or lose your $1,000 allotment. OK, fine, now what?

Well, what if you bet $1 on red or black each spin? What are your chances of winning $1,000 before you go broke? Let’s use the typical Las Vegas wheel that has both a zero and a double zero that result in a loss when you wager red or black.

The answer to how much you should bet simply rests on whether you want to maximize your chances of winning or stay longer. The bigger you bet, the better your chances of winning. The smaller you bet, the worse your chances of winning. Period.

I hear you mumbling. You’re saying, aren’t you Mike Caro? You’re supposed to be the logical, scientific gambling authority who preaches that it doesn’t matter how you manipulate your bets or choose your colors. If the odds are fixed against you, they’re against you, and there’s nothing you can do about it. Ah, so true.

But listen. I’m not saying you can whittle away at those odds against you. Those odds, given enough time to do their job, will extract almost exactly the percentage that the house expects. For every dollar you bet on red/black or odd/even at roulette, the house will eventually return 94¾ cents.

So, where’s the catch? Oh, the catch? The catch is that if you bet smaller amounts you will usually last longer before you either go broke or reach your goal. So? Isn’t it nice to last longer? Why does that make it better to bet bigger? Good question. The answer is that by lasting longer, you are actually betting more total money, even if your bets are smaller.

Let’s see some proof

To prove this powerful concept to you, I wrote a simple computer program that plays roulette. In the first trial, we took our $1,000 and bet either red or black each spin, determined to quit if we either lost all our money or doubled to $2,000. We bet $1 every single time, and we tried this experiment 100,000 times. Remember, we only imaged that we did, because we needed to use high-speed computer simulation to accomplish this.

The result of betting $1 each spin: We won 0.00% of the time (that’s right, we never won), and we lasted to make an average of 18,996 wagers per attempt to double our money. (This is very close to the number of wagers we would have mathematically expected, considering the house edge — but that’s another topic for another day.)

The result of betting $5 each spin: We won 0.00% of the time (again no wins!), and we lasted to make an average of 3,796 wagers per attempt to double our money.

If you’re very observant, you noticed something about that 3,796 figure. It’s about one-fifth as many bets as we got to make when we played for $1. But, since we were betting five times as much, the total about of money wagered in both cases was almost the same. Hmm… I wonder if this is always true. Does the total amount wagered always approximate the same thing, no matter how big or how small our bets? No! It was only because we never won and the bets were very small that the total wager was the same, as you’ll soon see.

The result of betting $25 each spin: We won 1.42% of the time (a little better now, we doubled our money about one time in 70), but we only lasted to make an average of 739 wagers per attempt. And, in fact, the total amount we wagered per attempt has fallen a little. But let’s really take chances…

The result of betting $100 each spin: We won 26.06% of the time (more than a quarter of the time), but we only lasted to make an average of 91.8 wagers per attempt to double our money. Our average total wagers are now down to less than $9,200 per attempt, whereas they were roughly twice that much previously.

It’s beginning to look like the more we wager on a single spin, the better our chances are to win — but only because we bet less total money. But, also, we don’t get to play as long on average. Let’s take it to an extreme.

The result of betting $500 each spin: We won 44.77% of the time, and we only lasted to make an average of just 3.99 wagers per attempt to double our money.

That’s just four wagers per attempt, on average, so — if you do things that way — you won’t get a lot of play before you have to go home.

And, of course, we could just bet it all on one spin, double or nothing, and we’d have our best shot possible. We’d win 18 out of 38 times on average or 47.37 percent of the time.

What it means

The point of today’s discussion is simple: If you have a predetermined goal for winning and a limited amount to lose, then…

If the odds are against you, you’re more likely to overcome those odds by betting big.

But, here’s the corollary — and you just need to think about what we’ve already discussed and put yourself on the side of the casino to understand this…

If the odds are in your favor, you’re more likely to win by betting smaller, even though your expected long-range profit for each dollar bet remains the same. And if the odds are against you, you’re more likely to win by betting bigger, even though your expected long-range loss for each dollar bet also remains the same.

Of course, you can bet too small and not maximize your profit opportunity. But, if you’re a skilled poker player, you do not want to jeopardize your bankroll unnecessarily. And you don’t want to let weak opponents playing limited funds escape. You’re more certain of victory if you play for reasonable limits, even though it may take more time to succeed.

This gambling concept is quite powerful, and it proves that — assuming your opponents have limited funds to lose — it is often your weakest poker foes who should want to raise the limits beyond what is comfortable, not you. — MC