Probability
How often do you roll 7 with two dice? Work it out — then roll a thousand times and see whether reality follows the theory.
Your numbers
More options: number of throws
Answer
Press «Roll» to throw. The bars show how often each result came up — the line is the theory.
How to work it out
- 1Possible outcomes in total: 6^2 = 36
- 2P(2) = 1/36 = 2.78 %
- 3P(3) = 2/36 = 5.56 %
- 4P(4) = 3/36 = 8.33 %
- 5P(5) = 4/36 = 11.11 %
- 6P(6) = 5/36 = 13.89 %
- 7P(7) = 6/36 = 16.67 %
- 8P(8) = 5/36 = 13.89 %
- 9P(9) = 4/36 = 11.11 %
- 10P(10) = 3/36 = 8.33 %
- 11P(11) = 2/36 = 5.56 %
- 12P(12) = 1/36 = 2.78 %
The formula
P(A and B) = P(A) · P(B) when A and B are independent
«And»: multiply along the branches of the tree diagram.
P(A or B) = P(A) + P(B) − P(A and B)
«Or»: add, and subtract the overlap so it is not counted twice.
P = favourable outcomes / possible outcomes
Applies when all outcomes are equally likely — as on a fair die.
K(n, r) = n! / (r! · (n − r)!)
How many ways can r items be chosen from n when the order does not matter (e.g. a lottery ticket)?
P(n, r) = n! / (n − r)!
The same, when the order matters (e.g. 1st, 2nd and 3rd place).
In depth: more numbers and “what if”
All outcomes and their probability
| Outcome | Number of ways | Fraction | Percent |
|---|---|---|---|
| 2 | 1 | 1/36 | 2.78 % |
| 3 | 2 | 1/18 | 5.56 % |
| 4 | 3 | 1/12 | 8.33 % |
| 5 | 4 | 1/9 | 11.11 % |
| 6 | 5 | 5/36 | 13.89 % |
| 7 | 6 | 1/6 | 16.67 % |
| 8 | 5 | 5/36 | 13.89 % |
| 9 | 4 | 1/9 | 11.11 % |
| 10 | 3 | 1/12 | 8.33 % |
| 11 | 2 | 1/18 | 5.56 % |
| 12 | 1 | 1/36 | 2.78 % |