For one, understanding dice probabilities will open up a world of calculating probabilities, which can be applied across a wide variety of applications and disciplines, especially when it comes to casino games. However, this guide is more focused on dice-based games such as craps and other traditional games that require dice totals to determine winners and losers.
Calculating Dice Probabilities – It’s Easier Than You Think
While it may sound rather complex and complicated (if one were just to go by the name of this guide – calculating dice probabilities), the truth is that it is considerably easier than you may believe. Like most skills that we learn in life, calculating dice probabilities begins with the basics and, once you’ve got those down, you can move up to more complex calculations.
The root of calculating dice probabilities begins with understanding a basic formula:
Dice probability is simply:
the number of possible results ÷ The number of desired results
In other words: If you were to desire a result of 6 (using just a single six-sided die), your probability would be:
1÷6 = 0.167
Which can also be expressed as a percentage, in this case: 16.7%
If one were to attempt to calculate the dice probability when employing two dice, the formula would be slightly different. Now we are dealing with a formula known as an Independent Probability Calculation.
So, your formula for calculating the independent results of each dice would be thus:
Probability of Both = Probability of result one (first dice) x Probability of result two (second dice)
In other words:
Let’s use our previous desired outcome of 6, only this time we want that result for both dice. Our calculation would then be:
1/6 x 1/6 = 1/36 = 1 ÷36 = 0.0278
Again, this can also be expressed as a probability percentage:
Starting with Just One Die Probabilities
If you are completely new to the concept of calculating the probabilities or outcomes of dice rolls in games, your best bet is to start with just a single die. This will make working out various calculations using the provided formula a lot easier to digest and to understand. It will also help to get you used to the art of calculating probability by looking at the number of possible outcomes compared to the results (the numbers you’re looking for) that are possible.
When we look at a single die, we see that there are six sides or six faces, numbered 1, 2, 3, 4, 5, and 6. This means that, at any given moment when the die is cast (thrown), there are six possible number results.
When applied practically, that is to say, when used in a game of chance, there will always be a specific number that you are after, over all other number facets on the die.
So, whichever number you decide is the one you are after, simply apply the formula previously covered to calculate your odds percentage:
Probability = the number of possible results against the number of desired results. So to summarise our example: Probability = 1 ÷6 = 0.167 (if 6 is your desired result. Otherwise simply replace this number with whatever number is the number that will give you the winning result).
In betting with dice, the probability is always defined between 0, which is no chance at all, and 1, which is the certainty of chance. The easiest way to express this is by multiplying your desired number by 100 to give you your probability as a percentage. In the case of our example (6) that would yield a probability percentage of 16.7 percent (chance).
Advancing to Two Dice Probabilities
Of course, most games that we all love to play usually use more than one die, the average being two. So it then becomes more useful to be able to calculate the probability percentages based on two dice, each with its own independent probability.
Now, it is important at this stage not to become flustered by the addition of another die. The formula is still simple enough to understand and to use on a practical level. In our example, we’ve used 6, the highest possible result on a standard six-sided gaming dice. When it comes to calculating the results of two (or more) dice, you then begin to move into the realm of independent probabilities. The reason for this is quite simple. The results of one die do not depend on the results of the other, which essentially means that you will need to calculate for each die separately.
Whenever you employ more than one die, you will have to deal with the separate chances or odds of each die in the roll producing the desired result that you are after. Unlike single die probability calculations, the golden rule for calculating independent probabilities is that you need to multiply the individual probabilities (each die in the roll) together to get your outcome.
With multiple dies in the roll, your formula for calculation is then modified slightly to accommodate for these changes.
While our single die formula contained only a single calculation string, your new calculation will be as follows: Probability of both die = probability of the outcome (result) of your first die times (x) the probability of the outcome of your second die.
To simplify this even further, we use fractions. Let’s say that, in our example, we’ve decided that the number 6 is optimal. Since there is a one in six chance of actually rolling a 6, your fractional representation will be 1/6. If we want both die to yield the optimal result of 6, our formula would look like this:
Probability = 1/6 X 1/6 = 1/36
To make this more useful to you, a final calculation is needed:
1/36 = 1 ÷ 36 or 0.0278
Again, this is easier to use in practical calculations when expressed as a percentage. So, if you’re looking for the probability percentage of rolling two6’s, that would be 2.78%.
What About Rolling Two Different Numbers from Two Dice?
Naturally, most games will not require prefect sixes for a win. You’ll often find that a variety of results are needed, depending on the game and what the win requirement is. Let’s say that your winning result requires a 5 and a 4. You can still apply the formula that we’ve looked at for rolling two dice with the same number result desired. However, since it doesn’t really matter which of the two dice produces the 4 and which produces the 5, you can apply the maximum 36 possible result matrix.
So, out of a total of 36 possible results, you’re looking for two specific numbers. By applying the following, slightly modified formula:
Probability = Number of possible results over the number of desired results = 2/36 or 0.0556
Again, we move this into a percentage value which helps to calculate probability more accurately:
You may have also noticed that this result actually gives you double probability over rolling two 6’s.
What About Calculating Total Score Resulting from Two (or more) Dice?
In some cases you may need to calculate the probability of getting a total score based on two or more dice. In this instance, we can still use our original formula:
Probability = The number of desired results ÷ the number of probable outcomes
However, for the purposes of determining totals, we’ll need to look a bit more into calculations based on different sides. While the same rules apply in terms of determining the total outcome based on multiplying the number of sides on your first die by the number of sides on your second die, counting the number of desired outcomes becomes slightly more tricky.
For example, if you were looking for a total of 4 from two dice, you’d have three possibilities to achieve this:
- Die One = 3 + Die Two = 1 Total = 4
- 2. Die One = 2 + Die Two = 2 Total = 4
- 3. Die One = 1 + Die Two = 3 Total = 4
So, to roll a pair of dice with the resulting total of 4, we now know that there are three possible paths to the desired result. We can thus apply the same 36 possible result matrix, giving us the following formula:
Probability= Number of desired results over the number of possible results giving you 3/36 or 0.0833
Again, we can express this as a percentage = 8.33%
Interestingly, the most common number to come up as a total when rolling two dice is 7. This is because there are six different ways to achieve this total result, with a probability of:
6 ÷ 36 = 0.167 or 16.7%
Dice Probabilities in Summary
When we talk of probabilities, we really mean the chance of something happening. Within the context of a standard six-sided die, your chances of rolling a 1 are just as good as your chances of rolling a 6. This is simply because each number represented has a 1 in 6 chance of coming up.
However, when you add a second die to the game, the odds of the dice yielding a result of 2snake eyes) are far less than the dice yielding a result of 7. The reason for this is also simple since the individual die can each only produce one possible result to create a total of two, that is 1 on each die (1+1). On the other end of the scale, producing a total of 7 from two die yields far more possibilities, including:
2 + 5
4 + 3
3 + 4
5 + 2
6 + 1
Ultimately, since there are always thirty-six possible results when rolling a pair of dice, it is always best to determine the total number of possible results based on your scenario. Another example, if you require a total result of 8 to win, you would need to determine how many ways this could be achieved. In the case of a total of 8, you’d have five possible combinations:
4 + 4
2 + 6
5 + 3
6 + 2
3 + 5
Finally, to determine the probability, divide the number of ways one could achieve the desired total result by the total number of possible outcomes, which will then give you your weighted probability. Dividing 5 by 36 would then give you your probability percentage of 13.89% (0.1389).