Use these calculators to find the probability of two events, solve for unknown probabilities of two events, compute the probability of a series of independent events, and find probabilities for a normal distribution.
Enter the probability of each independent event (a number between 0 and 1) to find all combined probabilities.
Fill in any two of the values below and the solver finds all the rest (assuming the events are independent). Leave the others blank.
Enter the probability of each event and how many times it occurs in the series.
Enter the mean, standard deviation, and a left and right bound to find the probability that a value falls in the range.
The probability calculator is a set of four tools for working with the probability of events. It can find every combined probability of two independent events, solve for unknown probabilities when you know just two values, compute the probability of a series of repeated independent events, and calculate probabilities under a normal distribution. Whether you are studying for a statistics exam, analyzing risk, working through a homework problem, or making a data-driven decision, these calculators handle the arithmetic so you can focus on interpreting the result.
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain), or equivalently between 0% and 100%. The calculators above apply the standard rules of probability to give you fast, accurate answers.
Probability is the branch of mathematics concerned with the likelihood of events. The probability of an event A, written P(A), is a number between 0 and 1. A probability of 0 means the event cannot happen; a probability of 1 means it is certain. For example, the probability of flipping a fair coin and getting heads is 0.5, and the probability of rolling a 7 on a single standard die is 0 because it is impossible.
For an experiment with equally likely outcomes, probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Rolling a 4 on a six-sided die has probability 1/6 because there is one favorable outcome out of six equally likely ones.
The first calculator takes the probability of two independent events, P(A) and P(B), and computes every combined probability. Two events are independent when the occurrence of one does not affect the probability of the other - like two separate coin flips. The calculator returns:
Example: With P(A) = 0.5 and P(B) = 0.4, the calculator gives P(A∩B) = 0.2, P(A∪B) = 0.7, P(AΔB) = 0.5, and P((A∪B)′) = 0.3.
The solver works backward. Instead of starting from P(A) and P(B), you provide any two of the eight related quantities - P(A), P(B), their complements, their intersection, union, symmetric difference, or the probability of neither - and the calculator derives all the others, assuming the two events are independent. This is handy when a problem gives you, say, P(A) and P(A∩B) and asks for P(B) or P(A∪B).
The solver relies on the same fundamental identities: the complement rule, the multiplication rule for independent events, the addition rule for unions, and the symmetric-difference formula. From two known values it reconstructs P(A) and P(B) and then computes the full set. Example: Given P(A) = 0.5 and P(A∩B) = 0.4, it finds P(B) = 0.8, P(A∪B) = 0.9, and P(AΔB) = 0.5.
This calculator handles repeated independent trials - for instance, the probability of an event happening several times in a row. If a single event A has probability P(A), then the probability of it occurring n times in a row is P(A)n, because independent probabilities multiply. The probability of it never occurring across n trials is (1 − P(A))n, and the probability of it occurring at least once is 1 − (1 − P(A))n.
The calculator extends this to two events, each with its own probability and its own number of repetitions, and reports the combined probabilities - both occurring, neither occurring, one but not the other, and more. Example: With A at 0.6 occurring 5 times and B at 0.3 occurring 3 times, the probability of A occurring all 5 times is 0.65 = 0.07776, and the probability of A occurring at least once is 1 − 0.45 = 0.98976.
This is the mathematics behind questions like "what is the chance of getting at least one six in four rolls of a die?" or "how likely is it that a system with several independent components all work?"
The normal distribution - the familiar symmetric "bell curve" - describes countless natural and statistical phenomena, from heights and test scores to measurement errors. It is defined by two parameters: the mean (μ), which locates the center, and the standard deviation (σ), which controls the spread. This calculator finds the probability that a normally distributed value falls between a left and a right bound by computing the area under the curve between them.
Internally it converts your bounds to z-scores - the number of standard deviations a value lies from the mean, z = (x − μ) / σ - and uses the standard normal cumulative distribution function to find the area. It reports the probability inside the range, the probability outside it, and the probabilities below the lower bound and above the upper bound.
Example: For a standard normal distribution (μ = 0, σ = 1), the probability of a value between −1 and 1 is 0.68269 - the well-known result that about 68% of values lie within one standard deviation of the mean. The calculator also produces a confidence-interval table showing the ranges that capture 80%, 90%, 95%, 99%, and other common confidence levels.
The probability that an event does not happen is one minus the probability that it does: P(A′) = 1 − P(A). Complements are often the easiest route to an answer - the probability of "at least one" success is usually found as one minus the probability of "none."
For independent events, the probability that both occur is the product of their individual probabilities: P(A∩B) = P(A) × P(B). For dependent events, you must use the conditional probability P(B | A), giving P(A∩B) = P(A) × P(B | A).
The probability that at least one of two events occurs is P(A∪B) = P(A) + P(B) − P(A∩B). Subtracting the intersection avoids double-counting the outcomes where both events happen. For mutually exclusive events (which cannot both occur), the intersection is zero and the rule simplifies to P(A) + P(B).
Understanding the relationship between events is essential to choosing the right formula:
Conditional probability, written P(A | B), is the probability of A given that B has occurred. It is defined as P(A | B) = P(A∩B) / P(B). Conditional probability is the foundation of Bayes' theorem, which lets you update probabilities as new evidence arrives and underpins fields from medical testing to machine learning. When two events are independent, conditioning makes no difference: P(A | B) = P(A).
P(A∩B) is the probability that both A and B occur (the intersection, "and"). P(A∪B) is the probability that at least one of them occurs (the union, "or"). The union is always greater than or equal to the intersection.
Use the complement: the probability of "at least one" equals 1 minus the probability of "none." For an event with probability p repeated n independent times, the chance of at least one occurrence is 1 − (1 − p)n. The series calculator above computes this directly.
Two events are independent when the outcome of one does not affect the probability of the other, such as separate coin flips. For independent events, the probability that both occur is simply the product of their probabilities.
This is the empirical rule: about 68% of values in a normal distribution fall within one standard deviation of the mean. Since a standard normal distribution has mean 0 and standard deviation 1, the range −1 to 1 captures roughly 68.27% of the area under the curve.
No. By definition, every probability is between 0 and 1 inclusive (0% to 100%). A value outside that range indicates a mistake in the calculation or inputs.
A z-score measures how many standard deviations a value lies from the mean: z = (x − μ) / σ. Converting a value to a z-score lets you look up or compute its probability using the standard normal distribution, regardless of the original mean and standard deviation.