I’ve taken a little bit of a sabbatical from blogging about card games lately, but I’m back today, and I’d like to write an article about odds and probability in card games and other games where luck is a factor. I know that’s different from our usual format here, but I thought it might be a fun change of pace. I also have a new review that I’ll be posting later today. By the way, if you hate math, you should just skip this post, because probability is ALL about math.
What Is Probability?
Probability is the mathematical study of chance. It’s how we measure the likelihood of certain events occurring. If you want to know how likely it is that you’ll be dealt an ace on the river, then you’ll get your answer using this branch of mathematics. If you want to know the likelihood of rain tomorrow, then this is your field. In fact, probability is used in almost all fields of human endeavor, from business to politics to science to education.
There are two kinds of probabilities. Some of these are uncertain, like when we try to predict how likely it is that the planet Earth will collide with an asteroid. They’re just estimates. The other kind, and the one we’re most interested in, is certain probability, where we know all the potential outcomes and the likelihood of each.
For example, if you ask someone to pick a number between 1 and 10 at random, and you want to know how likely it is that they’ll pick an even number, you know that the chances are 50/50. Half of those numbers are odd, and half of them are even, so in a truly random trial, the odds of getting an even number are the same as the odds of getting an odd number.
The easiest way to start thinking about probabilities is by thinking of them as fractions. The likelihood of something happen is a fraction where the number of ways the desired outcome is the numerator, and the number of total possible outcomes is the denominator. For example, if you want to calculate the chances of rolling a 6 on a single six-side die, you take the number of ways you can roll a 6 and divide it by the total number of potential outcomes. That probability becomes 1/6.
This can be expressed in multiple ways, including as a decimal, a percentage, or as odds. 0.1667 or 16.67% or 5 to 1 would be those expressions, respectively. You should have already learned how to calculate decimals and percentages in school. You might not have learned how to express such a number in odds format, though.
To express a probability in odds format, you take the number of ways something can’t happen and compare it to the number of ways something can happen. In this example, there are 5 ways to NOT roll a 6 and only 1 way to roll a 6. So the odds are 5 to 1.
This starts to matter in card games when you want to estimate how likely it is that the next card you’re dealt will help you or hurt you. For example, in a Texas holdem game, you might have 4 cards to a flush, and you want to know how likely it is that you’ll hit your card on the river.
There are 13 cards of each suit, but you already have 4 of them, so there are 9 of them left in the deck. 6 of the 52 cards in the deck are accounted for already–you have 2 of them in your hand, and there are 4 cards on the board. So there are 46 possibilities, and 9 of them will fill your hand, so your probability of hitting your flush is 9/46. That’s almost 20%, or close to 4 to 1.
How would this information help you? Suppose there’s $1000 in the pot, and you need to put $100 in the pot in order to stay in the hand. That’s a 10 to 1 payout if you win, compared to a 4 to 1 probability of winning. By understanding the odds, you’re able to estimate whether it’s mathematically correct to stay in the hand.
Sometimes you’ll want to estimate the likelihood of multiple things happening at the same time. In that case, you multiply the probabilities by each other. For example, if you’re playing an old-fashioned slot machine game, there are 10 symbols on each reel, and there are 3 reels. The probability of hitting a particular symbol, say a cherry, is 1/10 on the first reel. That chance is the same on the 2nd reel and on the 3rd reel, but if you want to know what the likelihood is for getting a cherry on all 3 of the reels at the same time, you’d multiply 1/10 x 1/10 X 1/10, and you’ll get an answer of 1/1000. That’s 999 to 1 on that particular symbol.
That kind of problem occurs when you’re calculating the odds of this happening AND of that happening. If you want to calculate the odds of this happening OR that happening, you add the probabilities together. Since there are 10 symbols, you theoretically have 10 different winning combinations. How do you figure out your chances of winning some combination? You add up the probabilities for each symbol:
1/1000 + 1/1000 + 1/1000 + 1/1000 + 1/1000 +1/1000 + 1/1000 + 1/1000 + 1/1000 + 1/1000 = 10/1000.
You can reduce that to 1/100, which converts to 99 to 1 odds on any single combination coming up.
The main thing to remember is that if the probability question includes the word “AND”, you multiple the probabilities. If the question includes the word “OR”, you add the probabilities.
I’d like to thank my cousin in Texas (who runs http://www.slotmachinemakers.com/) for that 2nd example regarding slot machines. I came up with the other examples myself.
Of course, this is just a beginner’s introduction to how probability works. You can find additional information about calculating probabilities on these pages:
- A Basic Probability Textbook Online – This one’s a little bit dry, but it’s detailed and accurate.
- A Beginner’s Primer on Probability from GnomeStew.com (See also Part 2 of that post.) By the way, if you’re into RPGs, Gnome Stew is one of the coolest blogs out there.
- Probability for Kids – I’ve found that with math-related topics in general, stuff that’s aimed at a younger audience is often more easily understood, especially by adults who have a little bit of math phobia.
- MathProblems.info – The author of this site specializes in probability, and he runs another site about casino gambling, too. The math problems on this particular site aren’t all probability-related, but it’s still a great resource.
Card Game Reviews Blog 
One of the fun things about probability is that you can often approach the same problem multiple ways. and get the same answer. Take your example about getting ANY winning combination on that slot machine for example. What we’re really interested in is the probability that all results come up the same, not what result it is, so you can also calculate it this way: The probability that the first wheel comes up as any result is 1 because we don’t care which result it is. The probability that the second result MATCHES the first result is 1/10 and the probability the third result matches the first two is 1/10. Thus the probability of getting 3 matching results of any type is 1 * 1/10 * 1/10 = 1/100 which is exactly the same result you came up with using a different method.
Hey, Matt–cool comment. You’re absolutely right about that, too. Probability, to my way of thinking, is the most fun kind of math, and also one of the most practical. Then again, I play a lot of games–card games, board games, etc., so probability might be less practical for someone else who has different interests from mine.