Dice Probability: Sums, Doubles and Common Mistakes
Understand fair dice with worked examples for sums, doubles and repeated rolls. Learn what changes the odds, and what a streak cannot tell you.
Start with the experiment
A probability answer is only as good as its assumptions. Here we use fair six-sided dice, numbered one to six, with independent rolls. Fair means each face has probability 1/6; independent means one result does not change another. The examples are arithmetic models, not a certification of any physical die or random generator.
Before rolling, write the exact event you want. “A six” could mean one die shows six, at least one of two dice shows six, or the total is six. Those are three different questions. Deciding after the result appears quietly changes the experiment.
Two dice have 36 ordered outcomes
Imagine one cyan die and one pink die. Each cyan face can pair with six pink faces, giving 6 × 6 = 36 equally likely ordered outcomes. A cyan one and pink six is a different outcome from cyan six and pink one, even though both total seven.
For totals 2 through 12, the outcome counts are 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1. Divide each count by 36. Seven therefore has probability 6/36 = 1/6; twelve has probability 1/36. The eleven possible totals are not equally likely because they contain different numbers of outcomes.
Doubles and at least one six
The doubles are (1,1), (2,2), (3,3), (4,4), (5,5) and (6,6). That is six outcomes, so doubles have probability 1/6. A particular double, such as two fours, has probability 1/36. Asking for any double multiplies the successful outcomes by six.
For at least one six, count the opposite event: neither die is six. Its probability is (5/6) × (5/6) = 25/36. Subtracting from one gives 11/36, about 30.56%. Adding 1/6 and 1/6 gives the wrong answer because the double-six outcome gets counted twice.
Repeated chances are not guarantees
Suppose you roll one die four times and want at least one six. The probability of no six is (5/6)^4 = 625/1296. The desired probability is 671/1296, about 51.77%. Four attempts do not give a 4/6 probability, because multiple sixes can occur in the same four-roll sequence.
The expected number of sixes in those four rolls is 4/6, or two thirds. An expectation is an average over many repeated experiments, not a possible result every single time. In one four-roll experiment you get a whole number of sixes, from zero to four.
What a streak does and does not mean
After five non-six results, an independent fair die still has probability 1/6 of showing six next. The previous rolls are already known; they do not create a debt that the die must repay. Likewise, a run of sixes does not make another six impossible.
A short sample can look uneven without proving bias. Record the number of rolls and the counting rule before comparing frequencies. If you roll again whenever a result feels wrong, your saved results no longer describe the original experiment. Keep every outcome when checking your own predictions.
Try a prediction challenge
Use the dice roller for a no-stakes experiment. Predict the number of sevens in 36 two-dice rolls: the expected count is six. Record all totals, then compare the observed count with six. Getting four or eight is not automatically a problem; random counts fluctuate.
Next, change only the number of faces. Two fair four-sided dice have 16 ordered outcomes, and a total of five has four of them, so its probability is 1/4. Rebuilding the outcome table works for unfamiliar dice too. The method matters more than memorizing one table.
Sources & further reading
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