Probability Calculator
Work out single-event, two-event and combination probabilities for dice, coins, cards and more, with the formula shown.
Calculate a probability
Example: rolling a 3 on a fair six-sided die has 1 favourable outcome out of 6 possible outcomes.
Your result
Two dice: probability of each total
Rolling two fair six-sided dice gives 36 equally likely outcomes. The table below was worked out with the same formula this calculator uses, and it shows why a total of 7 is the most likely result.
| Total | Ways to get it | Probability | Percentage |
|---|---|---|---|
| 2 | 1 | 1 / 36 | 2.78% |
| 3 | 2 | 2 / 36 | 5.56% |
| 4 | 3 | 3 / 36 | 8.33% |
| 5 | 4 | 4 / 36 | 11.11% |
| 6 | 5 | 5 / 36 | 13.89% |
| 7 | 6 | 6 / 36 | 16.67% |
| 8 | 5 | 5 / 36 | 13.89% |
| 9 | 4 | 4 / 36 | 11.11% |
| 10 | 3 | 3 / 36 | 8.33% |
| 11 | 2 | 2 / 36 | 5.56% |
| 12 | 1 | 1 / 36 | 2.78% |
How probability works
This free calculator works out probabilities for dice, coins, cards and combinations. It is for students, teachers, gamers and anyone checking a chance before making a decision. There is no sign-up, and everything runs in your browser, so nothing you enter is sent anywhere.
What is probability?
Probability is a number between 0 and 1 that measures how likely an event is. A probability of 0 means the event cannot happen, and a probability of 1 means it is certain. A fair coin has a probability of 0.5 of landing on heads. Probabilities are often shown as percentages, so 0.5 is the same as 50 percent.
Probabilities follow a few simple rules. All the probabilities of every possible outcome add up to 1, and the probability that an event does not happen is 1 minus the probability that it does happen.
How do you calculate the probability of a single event?
Divide the number of favourable outcomes by the total number of possible outcomes. For a fair six-sided die, there is one way to roll a 3 and six possible results, so the probability is 1 divided by 6, which is about 0.1667 or 16.67 percent. The same formula works for coins, cards and any event with equally likely outcomes.
The key is counting carefully. Drawing an ace from a standard deck has 4 favourable outcomes out of 52 cards, so the probability is 4 divided by 52, which is about 7.69 percent.
What is the difference between AND and OR probability?
AND probability is the chance that two events both happen, and you multiply their probabilities when the events are independent. OR probability is the chance that at least one of them happens, and you add the probabilities then subtract the chance of both. For two fair coins, both heads is 25 percent, while at least one head is 75 percent.
The subtraction in the OR formula matters, because the outcomes where both events happen would otherwise be counted twice. If the two events can never happen together, the overlap is zero and you simply add the probabilities.
What are combinations and how does nCr work?
A combination counts how many ways you can choose r items from n items when the order does not matter. The formula is n factorial divided by r factorial times n minus r factorial. Choosing 3 cards from 10 gives 120 combinations. The Powerball jackpot odds are about 1 in 292.2 million, according to the Multi-State Lottery Association in 2024.
Combinations differ from permutations, where the order does matter. There are 720 ways to arrange 3 cards in order from 10, but only 120 ways to choose them, because each set of 3 cards can be arranged in 6 orders.
What is the difference between probability and odds?
Probability describes the share of all outcomes that favour an event, while odds compare favourable outcomes with unfavourable ones. A probability of 1 in 6 is the same as odds of 1 to 5, because one outcome wins and five lose. Bookmakers quote odds, while scientists and statisticians usually quote probabilities, but both describe the same chance.
To convert, divide the probability by one minus the probability. A probability of 0.25 gives odds of 0.25 divided by 0.75, which is 1 to 3.
Frequently asked questions
Why is a total of 7 the most likely result with two dice?
There are six ways to make 7 (1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1), more than for any other total. That gives a probability of 6 out of 36, or about 16.67 percent.
Are coin flips really independent?
Yes, for a fair coin each flip is independent, which means the last result does not change the next one. A run of five heads does not make tails more likely on the next flip.
Can a probability be more than 100 percent?
No. Probability runs from 0 to 1, which is 0 to 100 percent. If your calculation gives more than 1, check that the favourable outcomes are not larger than the total outcomes.
What is the difference between combinations and permutations?
Combinations ignore order, while permutations count order. Choosing 3 lottery numbers is a combination, but a 3-digit lock code is a permutation, because 123 and 321 open different locks.
What do odds against mean?
Odds against are the reverse of odds in favour. If the odds in favour are 1 to 5, the odds against are 5 to 1, which means five outcomes lose for every one that wins.
Can a probability be exactly 0?
Yes, a probability of 0 means the event is impossible in the model you are using, such as rolling a 7 on a single six-sided die. In real life, a probability of 0 usually means the outcome is not in the set of possible results.
Is my data sent anywhere?
No. This calculator runs entirely in your browser. Nothing you enter is uploaded, stored or shared with anyone.
More statistics tools
Working with averages? Try the Mean Median Mode Calculator. Comparing two variables? Use the Correlation Coefficient Calculator. Ranking a score? Try the Percentile Calculator. For spread in more depth, see the Standard Deviation Calculator, and for standardised scores use the Z-Score Calculator, the P-Value Calculator and the Confidence Interval Calculator.
