Probability Calculator
Two events · Binomial trials · Normal distribution · Steps
🎲 Enter two event probabilities, a binomial setup, or a normal range to see every related probability with steps.
What Is Probability?
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be written as a decimal (0.25), a percentage (25%), or a fraction (1/4). This probability calculator covers the three tasks people search for most: combining two events, counting successes over repeated trials, and finding the chance that a normally distributed value falls in a range.
For a single event with equally likely outcomes, the basic probability formula is favorable outcomes divided by total outcomes. Rolling a 4 on a fair six sided die is 1 ÷ 6 = 0.1667. Drawing a heart from a standard 52 card deck is 13 ÷ 52 = 0.25. Getting heads on a fair coin is 1 ÷ 2 = 0.5.
The probability that an event does not happen is its complement. The complement is often the shortest route to an answer, especially for "at least one" questions.
How to Use the Probability Calculator
- Two events: enter P(A) and P(B), then choose the relationship. Independent events multiply, mutually exclusive events cannot overlap, and the dependent option takes the known P(A and B) so it can return conditional probabilities P(A | B) and P(B | A).
- Repeated trials: enter the success chance p for one trial, the number of trials n (up to 1,000), and the number of successes k. The result gives exactly k, at most k, at least k, and at least one success.
- Normal distribution: enter the mean, the standard deviation, and one or two bounds. Leave a bound blank for a one sided probability such as P(X ≤ 115).
Probabilities can be typed as decimals or percentages with the toggle; the values convert when you switch. Results update as you type and show the decimal, the percentage, and a "1 in N" figure. The step by step strip shows every substitution, so you can check the answer by hand.
Probability Formulas for Two Events
| Question | Formula |
|---|---|
| A and B, independent | P(A) × P(B) |
| A and B, any events | P(B) × P(A | B) |
| A or B | P(A) + P(B) − P(A and B) |
| Exactly one | P(A or B) − P(A and B) |
| Neither | 1 − P(A or B) |
| A given B | P(A and B) ÷ P(B) |
Independent and mutually exclusive are different ideas. Two coin flips are independent. Rolling a 1 and rolling a 6 on one die are mutually exclusive, so P(1 or 6) = 1/6 + 1/6 = 0.3333.
Conditional Probability and Dependent Events
Events are dependent when one changes the chance of the other, such as drawing cards without replacement. The conditional probability P(A | B) is the chance of A once B is known to have happened.
To test for independence, compare P(A and B) with P(A) × P(B). Here 0.4 × 0.3 = 0.12, which is not 0.2, so the events are dependent. The calculator rejects an overlap larger than either event or smaller than P(A) + P(B) − 1, because no real pair of events can produce it.
Without replacement example: the chance of two aces in two draws from a 52 card deck is 4/52 × 3/51 = 0.0045, because the second draw has one fewer ace and one fewer card.
Binomial Probability: Exactly, At Most, At Least
Repeated trials mode uses the binomial distribution: a fixed number of independent trials, two outcomes per trial, and the same success chance every time.
"At most 6" adds P(X = 0) through P(X = 6) = 0.8281. "At least 6" adds 6 through 10 = 0.3770. The mean is n × p = 5 and the standard deviation is √(n × p × (1 − p)) = 1.5811.
Normal Distribution Probability and Z Scores
For a normal distribution, convert each bound to a z score and read the cumulative probability Φ(z), the area under the curve to the left of z.
| Range | Probability inside | Outside |
|---|---|---|
| μ ± 1σ | 0.6827 | 0.3173 |
| μ ± 1.96σ | 0.9500 | 0.0500 |
| μ ± 2σ | 0.9545 | 0.0455 |
| μ ± 3σ | 0.9973 | 0.0027 |
For a single tail, P(X > 130) with the same mean and spread gives z = 2 and 1 − 0.9772 = 0.0228. Φ has no closed form, so the calculator uses a numerical approximation of the complementary error function with relative error below 1.2 × 10⁻⁷.
Odds, Interpretation, and Common Mistakes
Odds are not probability. Odds in favor compare successes with failures. A probability of 0.2 gives odds of 0.2 : 0.8, or 1 : 4 in favor and 4 : 1 against. To convert odds of a : b in favor back to probability, use a ÷ (a + b), so 1 : 4 becomes 1 ÷ 5 = 0.2.
Reading the result. A probability of 0.2051 means the outcome is expected about 205 times in 1,000 repetitions, or about 1 in 4.9. It does not promise that it will happen once in every five tries; short runs vary.
- Adding probabilities that overlap. P(A or B) is not P(A) + P(B) unless the events are mutually exclusive. Two events of 0.6 each would otherwise give 1.2, which is impossible.
- Multiplying dependent events. P(A) × P(B) only works for independent events. Cards drawn without replacement need the conditional probability.
- Mixing decimals and percents. Typing 50 when the toggle is set to decimal is rejected; switch the toggle to percent instead.
- Gambler's fallacy. After five tails, the next fair coin flip is still 0.5. Independent trials have no memory.
- Assuming normality. Normal mode is only accurate when the data are close to bell shaped. Skewed data such as incomes need another model.
Frequently Asked Questions
Sources & References
- NIST/SEMATECH. Engineering Statistics Handbook, 1.3.6.6.18 Binomial Distribution. itl.nist.gov
- NIST/SEMATECH. Engineering Statistics Handbook, 1.3.6.6.1 Normal Distribution. itl.nist.gov
- OpenStax. Introductory Statistics 2e, 3.2 Independent and Mutually Exclusive Events. openstax.org
- OpenStax. Introductory Statistics 2e, 3.3 Two Basic Rules of Probability. openstax.org
Results are for educational purposes. Probabilities describe long run frequencies under the stated assumptions and do not predict a single outcome. Methodology reviewed: 2026.