Probability Calculator

How to use the probability calculator

Two events A and B:

1. Enter P(A) and P(B) as decimals between 0 and 1. Enter 25% as 0.25.
2. Say how the events are related. Independent means one doesn’t change the chance of the other, like two coin flips. Mutually exclusive means they can’t both happen, like rolling a 2 and a 5 on one die. If neither fits, enter P(A and B) directly, or enter P(B given A), the chance of B once A has happened.
3. Read the results. The large figure is P(A or B), the chance that at least one happens. The cards show P(A and B), P(neither), P(exactly one), the two complements and both conditional probabilities. A note says whether the numbers make A and B independent.

Repeated trials: enter the chance of success on one try, the number of tries and a number of successes k. You get the chance of at least one success, of none, and of exactly, at least and at most k successes.

If the inputs are impossible together, such as mutually exclusive events whose probabilities add up to more than 1, the calculator says so instead of showing a number.

The rules behind the results

OpenStax’s Introductory Statistics 2e (chapter 3) sets out the rules this calculator uses:

Independent events are the special case where P(B | A) = P(B), so P(A and B) = P(A) × P(B). Mutually exclusive events are the special case where P(A and B) = 0, so the addition rule reduces to P(A) + P(B).

Everything else follows. P(neither) is 1 − P(A or B). P(exactly one) is P(A or B) − P(A and B). P(A only) is P(A) − P(A and B).

Worked examples from the textbook

Mutually exclusive (Example 3.14). Klaus can afford one vacation: New Zealand with probability 0.6 or Alaska with probability 0.35. He can’t take both, so P(A or B) = 0.6 + 0.35 = 0.95, and the chance he goes nowhere is 0.05.

Dependent events (Example 3.15). Carlos makes 65% of his shots, so P(A) = P(B) = 0.65 for two shots in a row. He shoots in streaks: given that he made the first, the chance he makes the second is 0.90. Then P(both) = 0.65 × 0.90 = 0.585, and P(either) = 0.65 + 0.65 − 0.585 = 0.715. Because 0.585 isn’t 0.65 × 0.65 = 0.4225, the shots aren’t independent. This is the calculator’s opening example.

Choosing which event is A (Example 3.17). Felicity takes a math class with probability 0.2 and a speech class with probability 0.65, and the chance of math given speech is 0.25. Because the known conditional is “math given speech,” enter speech as A (0.65) and math as B (0.2), with P(B | A) = 0.25. The calculator gives P(both) = 0.65 × 0.25 = 0.1625 and P(either) = 0.2 + 0.65 − 0.1625 = 0.6875.

Independent is not the same as mutually exclusive

The textbook stresses this point because students mix the two up. Mutually exclusive events can’t happen together, which means knowing one happened tells you the other didn’t. That is a strong dependence, not independence. Two events with nonzero probabilities can’t be both independent and mutually exclusive.

A coin flip and a die roll are independent: P(heads and a 6) = 0.5 × 1/6 = 1/12, about 8.33%. Rolling a 2 and rolling a 5 on the same die are mutually exclusive: P(both) = 0 and P(either) = 1/6 + 1/6 = 1/3.

At least once in n tries

The easiest way to find the chance of at least one success is through the complement. The chance of failing every time is (1 − p)ⁿ, so:

For at least one 6 in four rolls of a fair die, 1 − (5/6)⁴ = 1 − 0.4823 = 0.5177, a bit better than even. Many people guess 4 × 1/6 = 0.667, which adds probabilities of events that aren’t mutually exclusive. Adding works only when the events can’t overlap.

For exactly k successes, the calculator uses the binomial formula, C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ. It assumes what the textbook calls a binomial experiment: a fixed number of tries, two outcomes on each, and the same chance p on every independent try. Exactly 2 heads in 4 fair flips is C(4, 2) × 0.5⁴ = 6 ÷ 16 = 0.375.

A 1-in-100 chance per try needs about 69 tries before “at least once” passes 50%, and about 299 tries to pass 95%. Rare events become likely with enough attempts, but they don’t become certain.

Common mistakes

• Mixing up P(A | B) and P(B | A). The chance of a positive test given a condition is not the chance of the condition given a positive test. They are linked by P(A | B) = P(B | A) × P(A) ÷ P(B).
• Assuming independence. Multiplying P(A) × P(B) is only valid when one event doesn’t affect the other. Carlos’s streaky shooting is the textbook warning.
• Using percentages in a formula. 65% × 90% must be computed as 0.65 × 0.90, not 65 × 90.

Translating the wording

Probability questions hide the formula in everyday words. “Or” means at least one of the events, including both; “and” means both together; “not” or “doesn’t” points to the complement; “given,” “if” or “among those who” signals a conditional probability. “At least one” is usually easiest as 1 minus the chance of none, and “at most k” is the sum of the chances of 0 through k successes.

Odds are a different scale. Odds of 3 to 1 against an event mean a probability of 1 ÷ (3 + 1) = 0.25. To turn a probability into odds in favor, divide p by 1 − p: a probability of 0.25 is odds of 0.25 ÷ 0.75, or 1 to 3.

Percentages and decimals describe the same thing. This calculator takes decimals as inputs and shows results as percentages with the decimal underneath, so you can copy whichever form your assignment asks for. Keep at least four decimal places in intermediate steps; rounding 0.4823 to 0.48 before subtracting from 1 shifts the at-least-once answer by about 0.2 percentage points.

Sources & assumptions

How this calculator gets its numbers. Full method: our methodology.

• Complement P(not A) = 1 − P(A); addition rule P(A or B) = P(A) + P(B) − P(A and B); conditional P(A | B) = P(A and B) ÷ P(B).
• P(A and B) comes from your choice: P(A) × P(B) for independent events, 0 for mutually exclusive events, the value you enter, or P(A) × P(B | A).
• Inputs that can’t happen together, such as P(A and B) larger than P(A), are rejected rather than rounded.
• Repeated trials assume independent tries with the same chance p: at least once = 1 − (1 − p)ⁿ, exactly k = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ.

Sources

1. OpenStax (Rice University), Introductory Statistics 2e, 3.1 Terminology. Events, AND, OR, the complement (P(A) + P(A′) = 1) and conditional probability, P(A | B) = P(A AND B) ÷ P(B) with P(B) > 0.
2. OpenStax (Rice University), Introductory Statistics 2e, 3.2 Independent and Mutually Exclusive Events. Independent events: P(A | B) = P(A), P(B | A) = P(B) and P(A AND B) = P(A)P(B). Mutually exclusive events: P(A AND B) = 0. The two ideas are not the same.
3. OpenStax (Rice University), Introductory Statistics 2e, 3.3 Two Basic Rules of Probability. Multiplication rule P(A AND B) = P(B)P(A | B) and addition rule P(A OR B) = P(A) + P(B) − P(A AND B), with Example 3.14 (0.6 + 0.35 = 0.95), Example 3.15 (0.585 and 0.715) and Example 3.17 (0.1625 and 0.6875).
4. OpenStax (Rice University), Introductory Statistics 2e, 4.3 Binomial Distribution. A binomial experiment has a fixed number of independent trials n, two outcomes, and the same probability of success p on every trial.

Last reviewed November 2, 2026. Probabilities describe long-run frequencies; they don’t predict a single outcome.