When you walk into a casino, you have many different options for game play. Slot machines seem to be everywhere, with a few video poker machines mixed in. Also, groups of table games surround one or more pit areas, where players take their chances at popular games like craps, blackjack, and baccarat.
Outside of slot machines and roulette, almost every other game you find in a casino, including in the poker room uses either decks of playing cards or dice. The best thing about casino games that use dice or cards is you can learn how they work.
In order to do this you need to learn more about dice probabilities and card probabilities. Some players are afraid that learning about probabilities is too hard, but you only need to understand a few things. Below you’re going to learn a few simple mathematical truths about dice and playing cards.
Once you understand how probability works, you can make better decisions every time you play a dice or card game in the future. Don’t be afraid you won’t be able to understand. Everything you need to know is presented in an easy to understand manner.
Dice are used in craps and a few other casino games like Sic Bo. They’re also used in many popular board games. Once you understand how dice probability works, you can use your new knowledge at the casino and when you play board games.
When you consider a single die, it has six possible outcomes. The die has six sides, and each side has a different number. Dice are designed to be equally balanced, so in the long run each side is supposed to come up an equal number of times.
A single six sided die has an equal chance on every roll to land on one of the six sides. This means the probability of rolling a six is one out of every six rolls. The probability of rolling a one is the same.
If you convert this to a percentage chance, the probability of rolling a six is 16.67%. This number is rounded up for simplicity.
It’s pretty simple to understand the probability when you’re dealing with a single die. But when you add an additional die you have six times as many possible outcomes.
Instead of six possible outcomes, you have 36.
In order to understand how the probabilities work with two dice, you need to think about the two dice as separate things. To make it easy, let’s say you have one red die and one black die. When you roll both dice, the red die can land on one of six numbers, and the black die can land on one of six numbers.
This creates a total of 36 possible outcomes from rolling two dice. In the chart below, you can see the total of two dice added together, followed by the number of possible combinations, followed by the actual combinations. The first number in each combination is the red die, and the second is the black die.
|Total of Both Dice||Number of Possible Combinations||Actual Combinations|
|3||2 combinations||1-2, 2-1|
|4||3 combinations||1-3, 2-2, 3-1|
|5||4 combinations||1-4, 2-3, 3-2, 4-1|
|6||5 combinations||1-5, 2-4, 3-3, 4-2, 5-1|
|7||6 combinations||1-6, 2-5, 3-4, 4-3, 5-2, 6-1|
|8||5 combinations||2-6, 3-5, 4-4, 5-3, 6-2|
|9||4 Combinations||3-6, 4-5, 5-4, 6-3|
|10||3 Combinations||4-6, 5-5, 6-4|
|11||2 Combinations||5-6, 6-5|
Now that you know how many combinations each possible outcome has, and how the dice have to land to create the combination, you can determine the percentage chance of each outcome on any individual roll.
Notice that the chance of rolling a seven with two dice is the same percentage chance of rolling any single number when you roll one die. This means that when you roll two dice, you’re going to a roll a seven one out of every six rolls on average.
The probability of rolling a total of two, or snake eyes, with two dice is one out of 36 rolls. This is because there’s only one combination out of 36 that totals two. The red die and the black die both have to land on one.
You can use the number of combinations listed in the first chart to determine the probability of anything you need.
Here’s an example:
If you need to know the probability of a roll of two dice totaling six or eight, you simply add the number of combinations for each result. The six and eight both have five combinations, so you have a total of 10 combinations.
This means that 10 out of 36 times the result is either a six or an eight. You can convert this to a percentage by dividing the combinations, 10, by 36. In this case, the chance of rolling a six or eight is 27.78%.
Let’s look at how this information is useful in a real life situation. When you play craps, each round starts with a come out roll. When you bet on the pass line on the come out roll, if the roll is a two, three, or 12 you lose. If the roll is a seven or an 11, you win.
Here are the probabilities of losing and winning on the come out roll on a pas line wager:
Two and 12 each have one combination, and three has two combinations. This is a total of four combinations out of 36. This equals an 11.11% chance that you’re going to lose on the come out roll.
The seven has six combinations and the 11 has two, so the total number of combinations to win on the come out roll is eight. This equals a 22.22% chance to win on the come out roll.
In other words, you have twice the chance to win on the come out roll as you do to lose.
When the result of the come out roll is one of the other numbers, it sets a point. When a point is set, you need the shooter to roll the point before rolling a seven to win a pass line bet. The point can be any of the six remaining numbers; four, five, six, eight, nine, or 10.
The odds are always in the casino’s favor that a seven is rolled before a point. You can see why this is by looking at the first chart. Seven has six combinations, and the other six possibilities for the point have five, four, or three combinations.
You can use the same method I used to create the charts above to determine the probability of each outcome using three or more dice. The chart is bigger, but the basics are exactly the same.
Here’s a chart for three dice:
|Total of 3 Dice||Number of Possible Combinations|
Instead of 36 total combinations using two dice, using three dice has 216 possible combinations. To convert the number of combinations to a percentage, simply divide the number of combinations by 216.
Here’s an example:
The chance to roll a 10 is 27 out of 216. This equals 12.5%.
The chance to roll an 18 is one out of 216. This equals .46%.
Just like when you roll two dice, you can determine the chance of rolling two or more numbers with three dice by adding the combinations. The chance of rolling a 10 or 11 when rolling three dice is 54 out of 216, or 25%.
Blackjack, poker, and many other table games use one or more decks of playing cards. Even though many players don’t think about it, video poker machines all use playing cards also. Most games use a standard 52 card deck, but a few games use one or more jokers.
To keep things simple, this section only deals with the standard 52 card deck. But if you play a game using a joker, you can use the same methodology to determine the probabilities.
A deck of standard playing cards has 52 unique cards. This means the chance of the top card of a shuffled deck being any individual card is one out of 52, or 1.92%.
Each deck has four suits with each consisting of 13 cards. This means that the deck also has four cards of each of the 13 ranks. The suits are spades, hearts, clubs, and diamonds, and the ranks run from a low of two up to a high of ace. The ranks run from two to 10, followed by jack, queen, king, and ace.
Using what you know about how a deck of cards is constructed, you can determine the probability of many different things.
Here are a series of real world examples to show how you can use probabilities when gambling with cards:
You’re dealt a nine and a two, for a total of 11. You want to know if doubling down is a good play. The first thing to do is determine the probability of receiving a card that’s good for you against receiving one that isn’t.
Any of the face cards and 10’s gives you 21. A nine gives you a total of 20. An eight gives you 19, and a seven gives you 18. If you draw a six you have a total of 17, which is good, but not great. Any of the remaining cards, and ace, two, three, four, or five are not good cards.
Keep in mind that if you draw an ace to five, you still have a chance to win if the dealer busts, but these are still negative cards for you.
The math is simple now that you know which cards help you and which ones don’t. The cards you don’t want to draw total 20 out of the 52 possible cards. The cards that give you an 18 or higher total 28 out of 52. Four cards give you 17.
You don’t need to convert these numbers to percentages or odds beyond what you’ve already done, because it shows that 32 cards give you 17 or higher and only 20 give you 16 or lower. Even if you consider a total of 17 neutral, 28 cards give you 18 or higher and 20 give you 16 or lower.
If you fig deeper, you can see that 16 cards give you a total of 21, which is unbeatable. The worst possible outcome with 21 is a push, and most of the time the dealer doesn’t have 21.
You can use the same method to determine the possibilities for the dealer’s hand as you did for your hand, but you don’t need to do the math yourself. Just use a strategy card or chart, because the difficult math has been completed for you.
You’re playing Texas holdem and your hole cards are the ace and six of hearts. The flop has two hearts and a club. The turn is a spade. At this point you have four cards to a ace high heart flush and your single remaining opponent makes a bet.
At this point you’ve seen your two hole cards and four community cards. This leaves a total of 46 unseen cards. Nine of these cards complete your flush, and 37 of them don’t. This makes the odds 37 to 9 against hitting the flush. Looking at it another way, you have a 19.57% chance of hitting your flush on the river.
One more way to look at this is if you played this exact same hand 46 times, you hit your flush on average nine times, and don’t hit it 37 times.
By itself this information is only worth so much, but when you combine it with the amount in the pot and the amount of the bet you need to call, you can determine if it’s profitable or unprofitable in the long run to make the call. When you combine all of these things it’s called using pot odds, and this is one of the most valuable skills a poker player can develop.
You’re playing Jacks or Better and your first five cards are six of spades, seven of spades, eight of clubs, nine of spades, and ace of spades. You can either keep four cards to an outside straight or four cards to a flush. To determine the best play you need to consider the probability of improving each hand.
If you keep four cards to a straight, you need to draw one of the four fives or one of the four 10’s. Any other card creates a losing hand. This means you have eight cards out of the remaining 47 cards that complete your straight.
If you keep four to a flush, you need one of the remaining nine spades to complete a flush. But you also have an ace, so if you get one of the three remaining aces you have a pair of aces. This gives you 12 out of the remaining 47 cards that give you a paying hand.
You also need to consider the payouts for each hand. In this case the answer is clear; you should keep the four spades. A flush pays higher than a straight, and nine cards complete your flush and only eight complete a straight. The added benefit of possibly matching the ace just makes it better.
You can use what you know about a deck of cards to determine probabilities in many different situations. In the examples above you learned how using probability helped make the right play. But the thing I hope you learned is how to determine probabilities in any situation.
Start with the mathematical facts of a deck of cards that I mentioned at the beginning of the card section. Then consider what you need to know. Take the cards that have been played out of the deck or possibilities and then list the ones that complete what you need.
If you want to determine a percentage probability, divide the number of cards that help you by the total number of possible cards.
Here’s an example:
You’ve seen five cards, leaving 47 possible cards. 16 of the remaining cards help you, so the chance of the next card helping you is 34.04%.
In the long run, each possible unseen card will remain in the deck or another player’s hand the correct number of times. In a 10 handed Texas holdem game, each player starts with two cards, for total of 20. This leaves total of 32 cards in the deck that the dealer holds. But you can only see the two cards in your hand.
If you played a million hands starting with your two cards, each of the remaining 50 cards will be in each other possible location an equal number of times. Starting with the player to your left, let’s say they hold card one and two, the next player holds card three and four, and on through the players and then continue numbering the cards in the deck the dealer holds until you reach the bottom card, numbered 50.
Each number, one through 50, is a place for a card. Let’s say one of the cards that aren’t in your hand is the ace of clubs. It can be in any of the 50 spots. You don’t know which spot it occupies, but it’s in one of them. If you play the hand a million times, the ace of clubs will be the top card of the deck 20,000 times. It has the same chance of being the bottom card on the deck, or in any other position.
This is why you have to include every unseen card in your calculations. As soon as you see a card, you remove it from the list of cards in the remaining unseen spots.
Understanding how dice and card probabilities work instantly improves your gambling play. When you use this understanding to make betting decisions, it improves your results. Use the basic probability examples on this page to start thinking about how you can use probability every time you play a dice or card game in the casino and at home.