Understanding Probabilities When Playing Dice Games

Dice Probabilities      Dice games are probably the most widely played and well-known genre ever created. The concept of Dice games have been around for literally thousands of years, with ancient peoples sometime using shells and bones as Dice to add the element of controlled chance into their otherwise lackluster games. Also known to many as "Bones", Dice have been thrown by millions of people and have given birth to other games which are derived from their concept, most notably Dominoes.

      With so many Dice-based games available, we decided to create a collection of the various theory and strategy ideas associated with Dice for gamers to access whenever they needed them. Our dice probability charts are well organized and detail the probabilities that affect up to five dice when thrown simultaneously. We'll also detail the ease in which new Dice games can be created using the Chess and Poker Dot Com dice strategy guide by inventing one of our own based on the concepts of Dice Probability generated when multiple Dice interact. Let's have a look to see how dice interaction can be understood and utilized.

Alea jacta est: Dice Probability

Dice Interaction      The appeal of Dice games is rooted in the fact that when two or more Dice are thrown simultaneously, a beautiful interaction occurs. Before the initial throw the Dice are in fact separate entities. However, once multiple Dice are thrown, the totals they create (arrived at by adding all the values together) unbalance the games in which they are being used into various channels of chance and strategy. Based on the number of Dice thrown, certain totals become increasingly more likely than others. As more and more Dice are thrown together at once, the expected outcomes begin to fluctuate and alter the strategy that a player may use accordingly.

Probability is the number of times something occurs divided by the number of times it could occur.

      We'll take a look at the various combinations you will arrive at when rolling two, three, four and even five dice together, as well as some of the games that these outcomes affect. When researching the Dice statistics for this page, I could not find any one grouping of charts that showed all the numbers I was looking for. Logically, I then decided to create them myself. The following charts should be read from left to right, and they detail all of the possible Dice Counts, Sums and statistical percentages (probability) that they should occur. The following charts are based on the sums of all the Dice that are thrown. I call this the "Dice Count". For example, if you roll two Fives, your Dice Count would be 10 (5+5=10) and rolling a Three, Four and a Six would give you a 13 Count (3+4+6=13).

ChessandPoker.com Dice Probability Charts

      The top row of each chart (shaded black) show the various Dice Counts. In most cases, there are two Counts shown in each box. This means that the Sums and Percentages detailed below them apply to both of the counts listed. For example, 2 / 12 means that both Two Counts and Twelve Counts share the exact same probability of occurring. If only one Count is shown, then the percentages apply to only that certain Count.

      The middle row (shaded in white) show the exact number of combinations the Counts listed directly above them occur out of ALL THE POSSIBLE COMBINATIONS for that number of Dice. For example, in the Two Dice chart below there is a "3" listed in white below the 4 / 10 Counts. This means that when you're rolling two Dice, a Four and also a Ten both have 3 different combinations each out of the 36 total number of combinations possible between two Dice (there's more about this below). So there are three different ways to make a Four (1-3, 2-2, 3-1) and also three ways to make a Ten (4-6, 5-5, 6-4). And finally, the bottom row (shaded in blue) shows the number of possible combinations displayed as a percentage. Using the above 4 / 10 section for Two Dice, we see that each count has three different combinations or a 8.33% chance of occuring. This method is used for all of the charts in this guide.

2 or 12 3 or 11 4 or 10 5 or 9 6 or 8 7
1 2 3 4 5 6
2.78% 5.56% 8.33% 11.11% 13.89% 16.67%

Two Dice Probability (36 combinations)

3 or 18 4 or 17 5 or 16 6 or 15 7 or 14 8 or 13 9 or 12 10 or 11
1 3 6 10 15 21 25 27
.46% 1.39% 2.78% 4.63% 6.94% 9.72% 11.57% 12.50%

Three Dice Probability (216 combinations)

4/24 5/23 6/22 7/21 8/20 9/19 10/18 11/17 12/16 13/15 14
1 4 10 20 35 56 80 104 125 140 146
.08% .31% .77% 1.54% 2.70% 4.32% 6.17% 8.02% 9.65% 10.80% 11.27%

Four Dice Probability (1296 combinations)

5/30 6/29 7/28 8/27 9/26 10/25 11/24 12/23 13/22 14/21 15/20 16/19 17/18
1 5 15 35 70 126 205 305 420 540 651 735 780
.01% .06% .19% .45% .90% 1.62% 2.64% 3.92% 5.40% 6.94% 8.37% 9.45% 10.03%

Five Dice Probability (7776 combinations)

Understanding the Probability Charts

      As you can see, the Counts on the left side of the charts are much less likely to occur than those on the right hand side. The further right you go on a chart, the more likely the listed counts are to occur. For example, the right-most entry in the "Two Dice Probability" chart is the 7 Count. This means that the 7 Count is the most likely to occur, with a 16.67% chance of occurrence. So how do you utilize the raw data found in our charts? Here's an explanation with the oft-used two-dice game Craps. One of the first bets you can make in the game of Craps is the Pass Line Bet. This bet will win instantly for you when the shooter (the person throwing the dice) throws a 7 or 11 on their first throw. You will lose this bet if the shooter instead throws either a 2, 3 or 12 on the first throw. Let's use the Two Dice Probability Chart to determine the chances of each of those outcomes.

      The winning throws for the Pass Line bet are 7 or 11. Using the chart, we see that a 7 count has a 16.67% chance of occurring and the 11 count has about a 5.56% chance. That totals up to approximate winning chances of 22.23% The losing throws are 2 (2.78), 3 (5.56) or 12 (2.78) counts. So the losing chances equal 11.12%. We can see that concerning the Pass Line bet the shooter is almost twice as likely to roll a winning total than they are to roll a losing one. If this were the only bet and factor in the game it would be pretty simple, wouldn't it? Actually, there is much more to the game than just this as any Craps player will surely tell you. Using the chart again we can see that while the player is more likely to throw a winning 7 or 11 compared to a losing 2, 3 or 12, a whopping 66.66% (about 2/3 of the time) one of the other counts (4, 5, 6, 8, 9 or 10) should occur.

      Now we are beginning to see how the charts can be used to better understand the Dice games you are involved with. But what about the Dice games that don't already exist? Knowing the mechanics of what makes a dice game work should put you well on your way to creating games of your own. Want an easy 3 Dice game? Make a game where the players get huge rewards when they throw a 10 or 11 count. Since this is the most likely occurrence for three dice, more often than not they should come out pretty good. Need it a bit harder? Try a 4 dice game where you have to roll a 4, 5, 6, 22, 23 or 24 count to win a point. Even though that's 6 winning counts, the combined percentage chance of a winning count occurring is only 2.32%! Read through the following section that covers our own invented dice game, and note how the reward system is geared based on the likely outcomes of two dice.

Strung: Original Dice Game Invented by James Yates

      Let's explore our invented game called Strung to show how a game dependent on Dice sums might be constructed. The following table is brief rundown of the few rules of the game:

Number of Players: Two
Dice Required: Two Per Player
Wagering Devices: 50 Chips Per Player

Counts Actions for Each Count (The sum of both Dice)
4 thru 10 The lesser roll must contribute the difference in Chips into the Pot from their own stack. The numbers in this range are known as the Give Counts.
3 or 11 Player wins 5 chips out of the Pot (or all of them if the Pot has less than 5). These are both known as the Take Counts. They are both equal in strength and are very good Counts.
2 or 12 Player wins all chips currently in the Pot. These are known as the Strung Counts. They are both equal in strength and the best Counts you can get in the game.
  1. Both players roll their two dice simultaneously (but separate from each other).
  2. They then add their two dice together to arrive at their respective Dice Counts.
  3. Players gain or lose chips based on their own count compared to their opponents.
  4. The first player to capture all of their opponents Chips wins the game.
  5. In the case of both players rolling Strung Counts, regardless of whether or not they are 2 or 12 counts, both players must re-roll with no other action permitted.
  6. In the case of both players rolling Take Counts, regardless of whether or not they are 3 or 11 Counts, both players must also re-roll with no other action permitted.
  7. If one player rolls a Strung Count and the other a Take Count, the Strung Count "Trumps" the Take Count and wins the whole pot. The Take Count does not get to take their usual 5 chips in this situation.
  8. Finally, if both players roll matching Give Counts, they are simply required to keep re-rolling until one of the players must contribute to the Pot.

One Two Three Four Five Six

Gameplay

      To start the game, both players should each have a pair of dice and 50 wagering units each (in this example we used Poker chips). At the same time, each player rolls their two dice in front of them. They then each add up their Dice Count and compare it to the other players dice count. Chips are then won, lost or put into the Pot based on the chart above.

      As we know from the Two Dice Probability chart, there are 12 different Counts (sums) possible between two dice. The least likely of all the counts are the 2 and 12, which are tied with only one occurrence each. In this game, as in almost all other games of any genre, the rarer an occurrence is the more value it receives in the game. For example, in Poker the less likely a hand is to occur the higher it ranks. The Straight Flush is the most statistically unlikely hand in the game and is therefore the best hand you can have, followed by Four of a Kind which is the second least likely combination to occur. In most games LOW FREQUENCY EQUALS HIGH VALUE.

      So following this logic, the premium rolls in our game are the least likely of all the two dice combos. The two (1-1) and the twelve (6-6) counts make up only two combinations out of the total 36 combos for two dice. They are known as the Strung counts. If you roll a two or twelve and your opponent rolls anything other than a two or twelve themselves, you win all the chips in the current Pot. But if they also roll a Strung count, a stalemate occurs and both players must roll again with neither player winning any chips.

      The next unlikely group would be the three (1-2, 2-1) and eleven (5-6, 6-5) counts that are known as the Take counts. Therefore we have also given these rolls a special bonus. Whenever you roll either of these counts and your opponent doesn't also roll a Take count or Strung count, you get to take out five chips from the Pot into your stack. If your opponent had rolled a Strung count, unfortunately the Strung count would have trumped your second-best Take Count and they'd have won the whole Pot (you wouldn't even get to take out your five chips). So Take counts are the second least likely counts and beat everything except for the Strung counts.

      But how do you get any chips into the Pot in the first place? If neither player rolls a Strung or Take count, the player who has the lower count must contribute the difference in the counts to the Pot. So if you roll a 9 count and your opponent rolls a 5 count, they would have to put in four chips into the Pot. The difference between your higher count and their lower count is four (9 minus 5 equals 4), so that's the amount they have to cough up. Whether or not they'll be able to win it back on the next roll will depend entirely on the Dice. And this is as it should be!

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