Probability Calculator
Type chances as %, decimals or fractions and get every answer as you type, with the working. Work out two events, repeated tries, Bayes’ theorem, odds, dice, coins and cards.
Type 50%, 0.5, 1/2 or 1 in 2. A plain number above 1, like 40, is read as 40%.
Independent: one happening doesn’t change the chance of the other, like two coin flips.
Positive even though A is false (a false alarm).
As %, a decimal or a fraction: 25%, 0.25, 1/4.
Separate them with commas, spaces or new lines. Each one is taken as independent of the others.
How your probability is worked out
A probability is a number from 0 (never) to 1 (certain). Multiply for “and”, add for “or”, take from 1 for “not”.
- 1
Write as decimals
A 0.5 · B 0.4
- 2
Both happen
0.5 × 0.4 = 0.2
- 3
Either happens
0.5 + 0.4 − 0.2 = 0.7
- 4
Neither happens
1 − 0.7 = 0.3
Read more: the formulas, a worked example and quick tables
A probability is how likely something is, on a scale from 0 (it never happens) to 1 (it always happens). 0.25, 25% and 1/4 are the same chance written three ways, and so is "1 in 4". Almost every probability question comes down to three rules: multiply for "and", add for "or" (taking away any overlap), and take from 1 for "not".
The formula
not A: P(not A) = 1 − P(A)
A and B (independent): P(A and B) = P(A) × P(B)
A and B (any): P(A and B) = P(A) × P(B | A)
A or B: P(A or B) = P(A) + P(B) − P(A and B)
exactly one: P(A) + P(B) − 2 × P(A and B)
neither: 1 − P(A or B)
conditional: P(A | B) = P(A and B) ÷ P(B)
Bayes: P(A | B) = P(B | A) × P(A) ÷ P(B)
at least once in n tries: 1 − (1 − p)ⁿ
exactly k in n tries: C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
odds a : b against → probability = b ÷ (a + b)
normal: z = (x − mean) ÷ standard deviation
Worked example
Say A has a 50% chance and B has a 40% chance, and they don’t affect each other. As decimals they are 0.5 and 0.4.
- Both happen: 0.5 × 0.4 = 0.2 (20%)
- A or B (at least one): 0.5 + 0.4 − 0.2 = 0.7 (70%)
- Exactly one: 0.5 + 0.4 − 2 × 0.2 = 0.5 (50%)
- Neither: 1 − 0.7 = 0.3 (30%)
- Not A: 1 − 0.5 = 0.5, and not B: 1 − 0.4 = 0.6. Check: neither = 0.5 × 0.6 = 0.3.
More examples, one for each task
| Task | Example | Working |
|---|---|---|
| Repeated tries | Exactly 2 sixes in 10 rolls | C(10, 2) × (1/6)² × (5/6)⁸ = 45 × 1/36 × 0.2326 = 0.2907 (29.07%) |
| At least once | At least one six in 10 rolls | 1 − (5/6)¹⁰ = 1 − 0.1615 = 0.8385 (83.85%) |
| Bayes | 1% have it, test finds 90%, 9% false alarms | Of 10,000: 90 true positives, 891 false alarms. 90 ÷ 981 = 9.17% |
| Odds | 3 to 1 against | 1 ÷ (3 + 1) = 25%, decimal odds 4.0, American +300 |
| Dice | Total of 7 with 2 dice | 6 ways out of 36 = 1/6 (16.67%) |
| Cards | At least one ace in 5 cards | 1 − C(48, 5) ÷ C(52, 5) = 1 − 1,712,304 ÷ 2,598,960 = 34.12% |
| Several events | 50%, 30% and 1/4 | None: 0.5 × 0.7 × 0.75 = 0.2625, so at least one = 73.75% |
| Normal | Mean 100, SD 15, below 130 | z = (130 − 100) ÷ 15 = 2, P(Z < 2) = 97.72% |
Dice totals with two six-sided dice
| Total | Ways | Chance |
|---|---|---|
| 2 or 12 | 1 | 1/36 = 2.78% |
| 3 or 11 | 2 | 1/18 = 5.56% |
| 4 or 10 | 3 | 1/12 = 8.33% |
| 5 or 9 | 4 | 1/9 = 11.11% |
| 6 or 8 | 5 | 5/36 = 13.89% |
| 7 | 6 | 1/6 = 16.67% |
Independent, exclusive or linked?
- Independent: one doesn’t change the other, like two dice or two coin flips. Multiply for "and".
- Mutually exclusive: they can’t both happen, like rolling a 1 and a 6 on one die. "And" is 0 and "or" is a plain sum.
- Linked (dependent): one changes the other, like drawing two cards without putting the first back. Use P(A and B) = P(A) × P(B | A), or give the overlap in Advanced.
Tips
- "At least one" is almost always easiest as 1 − P(none).
- Chances don’t add up over tries: two tries at 50% give 75% for at least one success, not 100%.
- Past results don’t change a fair coin or die. After five heads, the next flip is still 50% heads.
- To count the ways to choose or arrange things (nCr and nPr), use the permutation and combination calculator.
The normal distribution uses a standard approximation of the error function, good to about 7 decimal places. Dice, coin and card answers are exact fractions.
Common questions
How do I work out the probability of A and B?
If A and B are independent (one doesn’t change the other), multiply them: P(A and B) = P(A) × P(B). With P(A) = 50% and P(B) = 40%, the chance of both is 0.5 × 0.4 = 0.2, or 20%. If they can’t both happen, P(A and B) is 0.
How do I work out the probability of A or B?
Add the two chances and take away the overlap so it isn’t counted twice: P(A or B) = P(A) + P(B) − P(A and B). For 50% and 40% independent events that is 0.5 + 0.4 − 0.2 = 0.7, or 70%. The chance of neither is 1 − 0.7 = 30%.
What is the chance of something happening at least once?
Work out the chance it never happens, then take it from 1. Rolling a six has a 1/6 chance, so missing it ten times in a row is (5/6)¹⁰ ≈ 0.1615. The chance of at least one six in ten rolls is 1 − 0.1615 ≈ 83.85%.
Why does a positive test not mean I have the condition?
Because false alarms from the many healthy people can outnumber the true cases. With 1% of people affected, a test that catches 90% of cases and gives 9% false positives finds 90 real cases and 891 false alarms in 10,000 people. Only 90 of the 981 positives, about 9.17%, really have it.
How do I turn odds into a probability?
For odds of a to b against, the probability is b ÷ (a + b). Odds of 3 to 1 against mean 1 win for every 3 losses, so the chance is 1 ÷ 4 = 25%. For odds in favour it is a ÷ (a + b). Decimal odds of 4.0 also mean 1 ÷ 4 = 25%.
What is the probability of rolling a 7 with two dice?
Two dice give 6 × 6 = 36 equally likely results. Six of them add up to 7: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1. So the chance is 6/36 = 1/6, about 16.67%. Seven is the most likely total with two dice.
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