The basics: favourable over possible
When all outcomes are equally likely, a probability is simply:
probability = number of favourable outcomes / total number of possible outcomes
The chance of rolling a six with one die is 1/6 ≈ 16.7%. The chance of an even number is 3/6 = 50%. A probability always lies between 0 (impossible) and 1 (certain), or between 0% and 100%. The chance that something does not happen is 1 minus the chance that it does.
At least once
What is the chance of rolling at least one six in four rolls? It is easiest to calculate the chance of no six at all and subtract that from 1:
chance of at least once = 1 − (1 − p)ⁿ
For four rolls: 1 − (5/6)⁴ ≈ 1 − 0.482 = 51.8%. You can't simply add the chances: with six rolls the chance of at least one six is not 100% but about 66.5%. The calculator also shows after how many tries your chance passes 50%.
Exactly k times: the binomial distribution
If you repeat something n times with the same chance p each time, the probability of exactly k successes is:
P(k) = (n choose k) × pᵏ × (1 − p)ⁿ⁻ᵏ
Flip a coin ten times, and the chance of exactly five heads is 24.6%. That is the most likely outcome, yet still less than a quarter. The calculator also gives the chance of at most and at least k times, and the expected number (n × p).
Combinations and permutations
How many ways are there to choose k items from n?
- Combinations (order doesn't matter): n choose k = n! / (k! × (n − k)!). A standard deck of 52 cards gives 2,598,960 different five-card poker hands, so the chance of being dealt one specific hand is 1 in 2,598,960.
- Permutations (order matters): n! / (n − k)!. How many different podiums (gold, silver, bronze) are possible with 8 runners? 8 × 7 × 6 = 336.
Independent events
With independent events, one outcome has no influence on another. A coin has no memory: after five heads in a row, the chance of heads on the sixth flip is still 50%. Believing that tails is now "due" is known as the gambler's fallacy. The chance that two independent events both happen is the product of their probabilities: rolling two sixes is 1/6 × 1/6 = 1/36.
Probability in everyday life
Probabilities are hard to judge by intuition. A 1% chance per try sounds small, but over 100 tries the chance of it happening at least once is about 63%. That matters for insurance, equipment failures and games. On the other hand, tiny probabilities such as a lottery jackpot are far smaller than we tend to imagine. To draw a random number yourself, use the random number generator; for the spread of data, see the standard deviation calculator.
Expected value
The expected value tells you what happens on average over many repetitions. Rolling a die 60 times, you can expect about 10 sixes (60 × 1/6), though any particular run of 60 rolls will rarely give exactly 10.
Frequently asked questions
How do you calculate a probability?
Divide the number of favourable outcomes by the total number of possible outcomes, provided all outcomes are equally likely.
How do I calculate the chance of at least once?
Use 1 − (1 − p)ⁿ, where p is the chance per try and n is the number of tries.
What does "n choose k" mean?
The number of ways to choose k items from n when order doesn't matter, also called the number of combinations.
What are the odds of a royal flush?
There are 4 royal flushes among 2,598,960 five-card hands, so the chance is 1 in 649,740 for a single five-card deal.
Can I add probabilities?
Only when the events are mutually exclusive, such as rolling a 1 or a 2 with one die. Otherwise multiply, or use the at-least-once formula.
What is the difference between probability and odds?
Probability compares favourable with total (1 in 6); odds compare favourable with unfavourable (1 to 5).
Last reviewed: 2026-10-06. Results are estimates for information only.