Probability Calculator
This probability calculator handles three scenarios in one place: the probability of a single event and its complement, the eight derived probabilities for two independent events (intersection, union, exclusive-or and more), and the area under the normal distribution curve between any two bounds. Enter your values and the results update instantly, with a worked-steps panel that shows exactly how each answer was reached.
Formula
Worked example
Two independent events: P(A) = 0.40, P(B) = 0.30. Then P(A and B) = 0.40 x 0.30 = 0.12, P(A or B) = 0.40 + 0.30 - 0.12 = 0.58, exactly one = 0.58 - 0.12 = 0.46, neither = 0.60 x 0.70 = 0.42.
What is probability?
Probability is a number between 0 and 1 that measures how likely an event is to occur. A probability of 0 means the event is impossible, a probability of 1 means it is certain, and any value in between represents a partial likelihood. The classical definition divides the number of favourable outcomes by the total number of equally likely outcomes: rolling an even number on a fair six-sided die has probability 3/6 = 0.5. The frequentist definition extends this to the long-run relative frequency of an event across many repeated trials: if you flip a fair coin 10,000 times, heads will appear close to 5,000 times. The Bayesian definition treats probability as a degree of belief, updated by evidence using Bayes' theorem.
How to use this calculator
Select "Single event" to enter one probability and get its complement. Select "Two independent events" to enter P(A) and P(B) and obtain all eight combined probabilities: complement of A, complement of B, intersection (both), union (at least one), exclusive-or (exactly one), and neither. These calculations assume the two events are statistically independent - the occurrence of one does not affect the other. Select "Normal distribution" to find the probability that a normally distributed variable falls between a lower and upper bound. Enter the mean and standard deviation of the distribution, then enter the interval endpoints; the calculator converts them to z-scores and integrates the normal curve between them. All modes display a worked-steps panel so you can follow every calculation.
Single-event probability and the complement rule
The complement of event A, written A' or not-A, is the set of all outcomes where A does not occur. Because one of these two things must always happen (either A occurs or it does not), P(A) + P(A') = 1. This gives the complement rule: P(A') = 1 - P(A). For example, if there is a 35% chance of rain, there is a 65% chance of no rain. The complement rule is one of the most frequently used shortcuts in probability because it is often easier to compute the probability that something does not happen than the probability that it does.
Two independent events: intersection, union, and more
When two events A and B are independent, knowing whether one occurred tells you nothing about the other. The joint probability P(A and B), also called the intersection, equals P(A) times P(B). The union P(A or B) is the probability that at least one of the events occurs; it equals P(A) + P(B) - P(A and B), where the intersection is subtracted to avoid double-counting outcomes where both events occur. The exclusive-or P(exactly one) equals the union minus the intersection. The probability that neither event occurs equals the product of the two complements: (1 - P(A)) times (1 - P(B)). If the events are NOT independent, you need the conditional probability P(A|B) to find P(A and B) = P(A|B) times P(B).
Normal distribution and z-scores
The normal distribution is a symmetric, bell-shaped curve fully defined by its mean (mu) and standard deviation (sigma). The probability that a normally distributed variable X falls between two values a and b is the area under the curve between a and b. To find this area, each boundary is converted to a z-score: z = (x - mu) / sigma. The z-score tells you how many standard deviations the value is from the mean. The area is then read from the standard normal cumulative distribution function (CDF). Key benchmarks: about 68.27% of values fall within one standard deviation of the mean, 95.45% within two, and 99.73% within three. This calculator performs the integration using a highly accurate rational approximation to the error function.
Key probability rules
| Rule | Formula | Notes |
|---|---|---|
| Complement | P(A') = 1 - P(A) | Always sums to 1 with P(A) |
| Intersection (independent) | P(A and B) = P(A) x P(B) | Only valid when A and B are independent |
| Union | P(A or B) = P(A) + P(B) - P(A and B) | Inclusion-exclusion principle |
| Exclusive-or | P(A xor B) = P(A) + P(B) - 2*P(A and B) | Exactly one of A or B, not both |
| Conditional | P(A|B) = P(A and B) / P(B) | Probability of A given B has occurred |
| Bayes' theorem | P(A|B) = P(B|A) * P(A) / P(B) | Updating beliefs with new evidence |
| Z-score (normal) | z = (x - mu) / sigma | Standardises to the standard normal |
| Normal CDF | P(X <= x) = Phi(z) | Phi = standard normal cumulative function |
Standard identities used in classical probability theory.
Frequently asked questions
What is the difference between mutually exclusive and independent events?
Mutually exclusive events cannot both occur at the same time - if one happens, the other cannot. For example, a single coin flip cannot produce both heads and tails. Independent events can both occur; the outcome of one simply does not affect the probability of the other. A fair coin flip and a separate die roll are independent. This calculator assumes independence for the two-event mode. Mutually exclusive events have P(A and B) = 0, so their union is just P(A) + P(B).
How do I calculate the probability of an event if I only know the odds?
Odds are expressed as "m to n" in favour of an event, meaning m favourable outcomes out of m + n total. The probability is m divided by (m + n). For example, odds of 3 to 1 in favour correspond to a probability of 3/(3+1) = 0.75. Conversely, if the probability is p, the odds in favour are p to (1 - p). Bookmakers often express odds differently (fractional, decimal, or American), each of which has its own conversion formula.
What is conditional probability and how does it differ from joint probability?
Joint probability P(A and B) is the probability that both A and B occur. Conditional probability P(A|B), read "probability of A given B," is the probability that A occurs knowing that B has already occurred. The relationship is P(A|B) = P(A and B) / P(B). When A and B are independent, P(A|B) = P(A) because knowing B gives you no information about A. Conditional probability is the foundation of Bayes' theorem, which tells you how to update your belief about A after observing B.
What does a z-score tell me?
A z-score measures how many standard deviations a value is above or below the mean of a normal distribution. A z-score of 0 is exactly at the mean, +1 is one standard deviation above, and -2 is two standard deviations below. Z-scores let you compare values from different normal distributions on the same scale, and they are used to look up probabilities from the standard normal table. This calculator converts your raw bounds to z-scores automatically.
How accurate is the normal distribution probability calculation?
This calculator uses the Abramowitz and Stegun rational approximation to the error function with a maximum absolute error of about 1.5 x 10^-7, which is sufficient for almost all practical applications. For very extreme z-scores (beyond about plus or minus 8), floating-point arithmetic limits precision, but those probabilities are already so close to 0 or 1 that the difference is negligible.
Can I use this for more than two events?
The two-event mode covers exactly two events. For three or more independent events, the principles extend naturally: the probability that all events occur is the product of their individual probabilities, and the probability that at least one occurs is one minus the probability that none occur. Many probability problems can be broken into pairs of events and solved step by step using the rules shown in the reference table above.