I've written about house odds vs true odds before, but not in so many words. You can read about the basics of probability on this site, and you can read about the house edge. But another way of looking at these concepts is by examining the difference between payoff odds and true odds.

Odds are just a way of stating a probability. That's just a mathematical way of looking at how likely it is that something is going to happen. All probabilities are ratios between events.

Here's an example:

You flip a coin. You want to know the probability that it's going to land on heads.

You have 2 possible outcomes—heads or tails.

Only one of them is the outcome you want the probability of.

½ is the probability of the coin landing on heads.

That probability can be expressed in multiple ways. A fraction is one way, but you could also convert that fraction into a percentage (50%) or a decimal (0.50).

But this post is about odds. How do you convert a probability into odds.

You just compare the number of ways it can't happen with the number of ways it can.

In the coin flip example, this means the odds are 1 to 1, or even odds.

Here's another example:

You're rolling a single six-sided die. You want to know the odds of rolling a 6.

You have 6 possibilities. Only one of them is a 6. The other 5 are NOT 6.

So the odds of rolling a 6 are 5 to 1.

One way of looking at house odds is in terms of the payoff for a bet. If the house pays off at odds less than your odds of winning the bet, over time, they're going to show a profit.

Here's an example:

You're at a casino which offers you a 4 to 1 payoff if you bet on a single number on a roll of a six-sided die. The odds of winning are 5 to 1.

You bet $1 every time, or a total of $6.

On 5 of those rolls, you lose a dollar. That's -$5.

On 1 of those you rolls, you win $4. (The bet pays off at 4 to 1, remember?)

$4 minus $5 results in a net loss of $1.

Of course, in the short run, you might win a few in a row or lose a few in a row. But in the long run, your actual results will start to resemble your theoretical results.

True odds are the actual odds of winning the bet. The difference between the true odds and the payoff odds is where the house gets its edge against the players. This is how the casino makes a profit on a consistent basis.

Here's another example:

You're playing roulette, and you place a single number bet on the number 2.

There are 38 numbers on a roulette wheel, each of which has an equal chance of coming up.

You have 37 ways to lose this bet and 1 way to win this bet.

The payoff for this bet is 35 to 1.

The true odds for this bet are 37 to 1.

Let's look at a theoretically perfect example where you place 38 roulette bets in a row. You win one of those bets and get $35. But you lose $37 on the other $37 bets, for a net loss of $2.

All casino games work in this manner. The casino pays off the bets at less than your odds of winning the bet. Over a large enough number of trials, the casino is almost certain to make a profit that's very predictable for each game.

Another phrase that gets thrown around a lot when discussing these kinds of issues is "payback percentage". This is a metric used to describe gambling machines. Table games almost always get talked about in terms of their "house edge", but video poker and slots are measured by their payback percentages.

The payback percentage is just the percentage of each bet that the casino expects to pay out in winnings over a long period of time. It's 100% minus the house edge.

Here's an example:

A casino at a bar in Las Vegas is programmed to have an 80% payback percentage. This means that every time you put a dollar in, you get 80 cents back.

But that's not literally true. You might win $2 on a spin, then win $20 on another spin, then lose 30 spins in a row.

But over time, the odds on that machine are programmed in such a manner that they'll add up to 80 cents on the dollar.

And the long run happens with slot machines faster than you think, actually. Most slot machine players make 600 spins per hour—sometimes more.

Imagine a casino with 1000 slot machines, all getting 600 spins per hour, 24 hours per day. That's over a million spins a day. At that rate, the casino expects to see actual results that resemble the expected results very closely indeed.

One of the most common uses for payoff odds vs. actual odds analysis happens in poker. It's not a casino game, and players play against each other. Poker players use a concept called "pot odds" to help them decide whether or not to call certain bets.

Here's an example:

You're playing Texas holdem, and you have 4 cards to a flush. The odds of filling your flush are around 2 to 1.

There's $10 in the pot, and it will cost you $2 to stay in the hand.

If you hit your flush and get paid off, you've made a 5 to 1 payoff.

Since your odds of winning are 2 to 1, this is a profitable situation.

On the other hand, if the pot only had $2 in it, and you had to put in $2 to stay in the hand, the situation wouldn't look nearly as good. You're getting an even money payoff if you win, but you don't have even odds of winning.

But this is one of the reasons that poker is a game where you can get an edge. You can fold in those situations where it's not profitable to put money in the pot. All you have to be able to do is analyze pot odds versus the odds of winning the hand and act accordingly.

House odds and true odds aren't hard concepts to understand. They're fundamental to gambling games, though. Casinos make their profits by paying bets off at less than the odds of winning. This holds true for every bet in the casino.

Understanding how odds work is a fundamental step in becoming an educated gambler.