The Power of Positive Expectation Gambling

Hand Flipping a Quarter With a Money Background

You’ve heard of compound interest, right?

And you’ve heard that Einstein said that compound interest was the most powerful force in the universe?

Well, the expectation in gambling is like compound interest on steroids.

In this post, I explain the differences between negative expectation and positive expectation gambling and what that means for your finances in the long run.

Every Bet Has an Expected Value, Even If That Value Is Zero

If I bet you a quarter that you’d flip a coin and it would land on heads, you’d have a 50% probability of winning. So would I.

In the short run – on that single bet – one of us would win, and the other one would lose.

If we made that bet twice in a row, one of the following would happen:

  1. I’d win twice.
  2. You’d win twice.
  3. You’d win the first toss, and I’d win the second.
  4. I’d win the first toss, and you’d win the second.

But, in the long, over the course of hundreds or thousands of coin tosses, if you and I were betting even money, we’d both break even.

That’s a bet with a zero expected value.

Even money just means that each of us wins or loses the same amount.

But if we change that equation to where I win 50 cents when I win, but you only win a quarter when you win, I have a positive expected value, and you have a negative expected value.

This is how almost all gambling works, by the way. Someone almost always has a mathematical advantage over someone else. This is how casinos and sports books stay in business, in fact.

But how big a deal is this mathematical expectation?

The Concept of the House Edge

The house edge is a statistical way to measure how much, on average, over the long run, how much you expect to lose per average on a bet against the casino.

In the coin tossing example earlier, it’s easy to calculate.

Let’s assume that you’re risking $200 to win $100, and you have statistically perfect expected results over the two coin tosses. You win $100, but you lose $200, for a net loss of $100. Over two coin tosses, that’s an average of $50 lost per bet. Since $50 is 50% of $100, we’d say that this bet has a 50% house edge.

You’d be a fool to make this bet, obviously, but this is the mathematical principle that applies to all bets in the casino.

What If You Have an Edge Over the Casino?

In most cases, you’ll never have a mathematical edge over the casino. Most games just don’t have an opportunity to use any kind of strategy to get such a mathematical edge.

But if you did find a game where you could get an edge, you could apply the rule of 72 to your edge to figure out how long it would take you to double your bankroll if you reinvest all your winnings over time.

After all, the edge you have over the casino is just return on investment, and the rule of 72 applies to return on investment.

Only, instead of looking at your return on investment on an annual basis, you’re looking at your return on investment on a per bet basis.

The rule of 72 suggests that if you divide your annual return on investment into 72, you’ll get the number of years it will take to double your money. For example, if you’re earning a 12% return on your money annually, you’ll double your money every six years.

That’s compound interest in action, folks.

Let’s say that you find a situation in a casino where you can get a 1% edge over the casino. Applying the rule of 72 to this, you’d think it would take 72 years to double your money.

But since you’re seeing that 1% return on average on every bet, it only takes 72 bets for you to double your money.

Where Can You Get a 1% Edge Over the Casino?

The surest ways I know of to get a 1% edge when gambling are to count cards in blackjack, play poker at an expert level, and to handicap sports better than the sportsbook.

If you want to figure out how to maximize your return on investment, you want to start thinking about how many bets you can get in per hour.

Person at a Poker Table With a Large Stack of Chips

If you play blackjack, you can get in more bets per hour than you can in poker or sports betting. The number of hands per hour you’ll get in blackjack varies based on how many players are at the table. If you’re heads-up with the dealer, you’ll obviously get more hands per hour than you would if you’re at a full table with six other blackjack players.

I’ve seen various estimates. I’ve seen some writers contend that you can play 350 hands per hour heads-up with the dealer, but I’ve seen other writers use a number of 200 hands per hour. I think the difference lies with how many hands you play.

If you’re the only player at the table, you can play two hands at a time. If you’re doing that, the 350 hands per hour figure makes sense.

On the other hand, if you’re at a table with six other players, you’re looking at closer to 50 or 60 hands per hour.

Does This Mean I Can Get Rich Counting Cards?

Sort of, yeah.

Let’s say you start by making $5 per hand bets with a 1% edge. If things go well, you should be able to double your casino bankroll within an hour or two.

At that point, you can double the size of your bet to $10 per hand.

It doesn’t take long when you’re doubling your average bet size to have a huge bankroll.

Gambling isn’t a sure thing. You’re not investing in bonds here. You could hit a string of bad luck. In that respect, gambling with a positive expectation resembles investing in the stock market. No stock is a sure thing.

You can even go broke with a positive expectation if you hit a long enough unlucky streak.

The trick is having a big enough bankroll to withstand the vagaries of luck. You want to minimize your risk of ruin.

Most card counters think in terms of having a certain number of betting units. With $2000 or so, you can play for $5 per hand in blackjack with a minimal risk of going broke.

How Does One Get This Edge in Blackjack?

Counting cards isn’t as hard as you think. It works because the deck of cards has a memory of sorts – once a card has been dealt, it can’t be dealt again until the deck is reshuffled. This changes the probabilities of almost everything to do with the game.

And since the cards are arranged randomly, sometimes the deck will be relatively rich in cards which are advantageous to the player, while other times, the deck will be relatively rich in cards which are advantageous to the casino.

Which cards are these?

Since a natural – a 2-card hand totaling 21 – pays off at 3 to 2, it’s advantageous to have a better probability of getting a natural.

And since the only cards that can form such a hand are the tens and aces, a deck with a relatively high number of tens and aces in it is advantageous to the player.

Closeup of Multiple Hands of Blackjack

When you can identify such a situation, you raise the size of your bets. This is how you get your edge in blackjack when counting cards.

The better the count, the more you bet.

It’s as simple as subtracting 1 from the running count every time you see a 10 or an ace and adding 1 to the running count every time you see a 2, 3, 4, 5, or 6.

When the count is zero or negative, bet the minimum.

As the count gets higher, raise the size of your bets in proportion to the count.

It’s a little more involved than this, but not by much.

If All This Is True, Why Doesn’t Everyone Get Rich Gambling?

Not everyone wants to make their living playing casino games. Some people – believe it or not – don’t enjoy playing blackjack. And some people who enjoy blackjack don’t enjoy the rigor of counting cards.

Other people want to earn their livings by contributing to society. You can’t blame them for that.

Also, not everyone has the determination, discipline, and focus required to pull off counting cards in a casino.

Don’t forget. The casinos frown on card counting, and if they catch you, they’ll stop you. It’s not illegal to count cards, but casinos can ban you from their blackjack tables. In fact, they can ban you from the premises entirely if they decide that’s what’s needed.


The power of positive expectation gambling should be obvious. Casinos make positive expectation bets almost 100% of the time, and look how much money they have.

If you can learn how to make positive expectation bets, you can double your money multiple times and get rich, too.

And counting cards is just one example of how to do it.