What Is Probability?

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Ask a casino gambler what probability means, and they’ll probably say something like “the chance of something happening.” That’s true, but only because chance is another word for probability. Other ways of saying the same thing are the odds, or the likelihood.

Sadly, just rattling off synonyms doesn’t help us understand something.

If you want to have the most fun possible and stay safe while playing in a casino, it’s important to master the basics of casino probability.

If you asked a scientist, they might give you a different definition of probability:

The expected frequency of an event over a large number of trials.

That might sound a bit technical, but it’s a simple idea.

Imagine that you can try the same experiment over and over again, lots of times. The probability is how often the result you’re looking for happens (on average) compared to the total number of tries.

Different Ways of Expressing Probability

You’ll hear people talk about probability in a few different ways. Experienced gamblers are good at switching between these in their heads.


You might be most used to seeing probability written as a percent chance. “Percent” means “per hundred” in Latin, so a 25% chance means that if we try something enough times, we’ll find that we succeed about 25 times for every 100 attempts.

When your weather forecaster says there’s a 25% chance of rain in your area, they mean that if you had 100 days that looked just like this, your picnic would get rained on about 25 times.


We don’t have to use 100 as our basis for comparison, though. You could express the same 25% probability by saying it’s one in four. 

They’re the same thing because 1/4 = 25/100.

You can use any other fraction the same way, like two in five or 13 in 27. This way of talking about probability is especially useful when we know how many different possibilities there are.

For instance, there are 52 cards in a deck and only four Aces. So the probability of pulling an Ace from a full deck is 4 in 52, which is the same as 1 in 13.


A third way of saying there’s a 25% chance is that the odds are three to one (sometimes written 3-1 or 3:1).

Wait, what? You might have expected that one in four would be the same as four to one. But when we’re talking about odds using the “X-to-One” format, we’re comparing successes to failures, not to the total number of tries.

A one-in-four chance means that for every four tries, we’ll have one success. But that means we’ll also have three failures. Three failures to one success means three-to-one odds.

That style of odds is the format casinos use most often. You’ll understand why when we get to the lesson on Odds and Payouts.

"3 to 1," "1 in 4" and "25%" are three ways of expressing the same probability.

Probability vs. Frequency

That technical definition we gave up top included the word “frequency,” that is, how often something happens.

However, two things happening with the same frequency aren’t necessarily following the same pattern. One could be totally regular and predictable, like a metronome. The other could be completely random and unpredictable. It could also be somewhere in between those extremes.

Here we have some jelly beans in an orderly sequence. You can clearly see that one in three is a green jelly bean, while the rest are red.

A uniform frequency creates a predictable pattern.

Assuming this sequence continues, we wouldn’t say there’s “a one-in-three chance” the next jelly bean is green. Because we know the pattern and we’ve just seen two red jelly beans in a row, we’d say there’s a 100% chance the next jelly bean is green.

But casino games are based on randomness. Imagine we take those same jelly beans and mix them up in a bag. The frequency of green jelly beans is still the same, but now it’s not predictable when I’ll pull one out. Now it really is a one-in-three probability because we’ve jumbled the sequence.

In other words, the probability becomes the same as the frequency after you add randomness.

The Law of Large Numbers and the Gambler’s Fallacy

There are two very important things to keep in mind when it comes to probability and frequency. One is what mathematicians call “the Law of Large Numbers.” The other is kind of its opposite, a mental trap that you’ll want to avoid, called “The Gambler’s Fallacy.”

The Law of Large Numbers

When our jelly beans were all lined up, we were guaranteed exactly one green bean for every two red ones: red, red, green, red, red, green…

Once they’re mixed up in the bag, that’s no longer true. Our chances of pulling a green bean might be one in three, but we could pretty easily get two or three in a row.

By the same token, we might get a dozen or more red beans without any greens at all.

But the more tries we take, the less extreme the results are likely to be.

We’re 11% likely to get two greens in a row, for instance, but only 0.1% likely to get six in a row. (To learn more about how to calculate those numbers, see our lesson on Combining Probabilities).

That’s true no matter how big the bag is, and it’s also true for things like rolling dice or flipping coins.

In a nutshell, the Law of Large Numbers says:

The more times you try something, the closer you’re going to get to the frequency you expect. 

Random probabilities can lead to streaks, but luck will start to even out in the long run. There’s a catch, though.

The Gambler’s Fallacy

A fallacy is a lie we tell ourselves or something that seems obvious but is actually wrong. 

The Gambler’s Fallacy is the expectation that luck “tries” to even itself out in the short term. Put another way, it’s like believing there’s a magic force pulling probabilities back toward the frequency you expect.

It’s that voice in your head that says your coin flipped heads three times in a row, so tails “must be coming.”

That’s not true for most things. It’s not true for dice, coin flips or the roulette wheel. (It is true for a deck of cards – see our article on Card Counting in Blackjack –  but only if you leave out the cards you’ve already pulled. As soon as you reshuffle the deck, you reset the probabilities.)

The important thing to remember about the Law of Large Numbers is that it’s only the frequency that evens out, not the number of hits or misses. And that’s only because the number of tries keeps going up.A player's average deviation from the expected number of wins increases over time, even if it gets smaller as a proportion of the number of attempts.

For instance, imagine flipping a coin. Maybe it goes like this:

  • After 10 flips: 7 heads, 3 tails (70% heads)
  • After 100 flips: 55 heads, 45 tails (55% heads)
  • After 1000 flips: 520 heads, 480 tails (52% heads)

As you can see, the percentage (frequency) of heads is getting closer and closer to 50%. But imagine a bettor gambling on tails. He was only down by four bets after 10 flips, but he’s down by 40 by the 1000th flip.

In other words, the fact that the frequency evens out isn’t a guarantee that you’re going to “catch up.” The average size of your net winnings or losses (“deviation”) will tend to keep going up, just not as fast as the number of tries.


Casino School: Probability

This has been Probability Lesson 1 of our four-part Casino School series. Next up is Probability Lesson 2: Game Odds Vs. Payout Odds.

About the Author

Alex Weldon

Alex Weldon

Alex Weldon is an online gambling industry analyst with nearly ten years of experience. He currently serves as Casino News Managing Editor for Bonus.com, part of the Catena Media Network. Other gambling news sites he has contributed to include PlayUSA and Online Poker Report, and his writing has been cited in The Atlantic.
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