Probability Calculator

Probability measures the likelihood of an event on a scale from 0 (impossible) to 1 (certain). The probability of drawing an ace from a shuffled deck is 4/52 ≈ 0.077 or about 7.7%. The complement (drawing a non-ace) is 1 minus that, or about 92.3%. Combined event rules extend this to two events. For independent events (where one outcome does not affect the other), P(A and B) = P(A) × P(B). For mutually exclusive events (where A and B cannot both occur), P(A and B) = 0 and P(A or B) = P(A) + P(B). This calculator covers all three common cases: basic probability with its complement, the intersection of two events, and the union of two events.

• Risk assessment: If two independent systems each have a 5% failure rate, P(at least one fails) = P(A or B) = 1 - (0.95 × 0.95) = 9.75%. Use the "or" mode with P(A)=P(B)=0.05.
• Card and dice games: Rolling two dice: P(getting a 6 on die 1 or die 2) with independent rolls of 1/6 each. The or calculation gives the combined probability.
• Medical testing: If a test has 95% sensitivity (P(positive | disease) = 0.95) and the disease has 1% prevalence, use the basic mode to understand what a positive result means.
• Weather and planning: P(rain) = 0.4 on Saturday, P(rain) = 0.3 on Sunday, independent days. P(rain at least one day) uses the "or" formula.

1. 1Select the calculation type: basic, and (intersection), or or (union).
2. 2Enter P(A) as a decimal between 0 and 1. For example, 0.3 means a 30% chance.
3. 3For "and" and "or" modes, also enter P(B) and specify whether the events are independent or mutually exclusive.
4. 4The calculator applies the appropriate probability rule.
5. 5Results are displayed as percentage, decimal, and approximate fraction.

• Adding probabilities without subtracting the overlap: P(A or B) is not simply P(A) + P(B) unless the events are mutually exclusive. For independent events, the intersection P(A and B) must be subtracted to avoid counting it twice.
• Confusing independent with mutually exclusive: Independent events can both occur; mutually exclusive events cannot. Coin flips are independent. Getting heads and tails on the same flip are mutually exclusive.
• Entering probabilities greater than 1: Probabilities must be between 0 and 1. If you have a percentage (like 30%), divide by 100 first to get 0.3 before entering it.
• Assuming all real-world events are independent: Many practical events are not independent. Weather on consecutive days, stock returns in a downturn, and related system failures are all correlated. Check independence assumptions before using the multiplication rule.

• Use the complement for "at least one" problems: P(at least one event in n trials) = 1 − P(none). For example, P(at least one head in 3 flips) = 1 − (0.5)^3 = 0.875. Often easier than summing individual cases.
• Express results in all three forms: Probabilities as fractions (like 1/6 or 3/8) are intuitive for card and dice problems. Percentages work for risk and weather. Decimals are needed for formula chaining.
• Check with the sure-thing rule: P(A or not A) must equal 1. After any calculation, confirm that the event probability plus its complement adds up to 1. If not, you have an error somewhere.
• For multiple independent events, multiply: P(A and B and C) = P(A) × P(B) × P(C) when all three are independent. Extending the and rule to three or more events is straightforward multiplication.