Probability Calculator

Probability of Two Events

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can also be written as a percentage, so a probability of 0.25 is a 25% chance. When every outcome is equally likely, the probability of an event is the number of outcomes in the event divided by the total number of outcomes: a fair die shows an even number with probability 3/6 = 0.5.

The calculators above assume that A and B are independent, meaning that whether one happens has no effect on the chance of the other. Two coin flips, or a coin flip and a die roll, are independent. Drawing two cards from a deck without replacing the first is not, because the first draw changes what is left. Independence is what makes the probability of both events simply the product of their individual probabilities. The sections below describe each combination the calculator reports.

Complement of A and B

The complement of an event is everything that can happen except that event. It is written A' (or Aᶜ, or Ā) and read as not A. Since an event either happens or it does not, the two probabilities always add up to one: P(A') = 1 − P(A).

Complements often make hard questions easy. The chance of rolling at least one six in four rolls of a die is awkward to count directly, but the chance of rolling no sixes is (5/6)⁴ ≈ 0.482, so the chance of at least one six is 1 − 0.482 ≈ 0.518. The repeated-events calculator uses the same trick for its at-least-once results.

Intersection of A and B

The intersection of A and B, written A∩B, is the event that both happen. In a Venn diagram it is the overlapping region of the two circles. For independent events, its probability is the product of the two probabilities: P(A∩B) = P(A) × P(B). With P(A) = 0.5 and P(B) = 0.4, the chance of both is 0.2.

Suppose you flip a fair coin and roll a fair die. The chance of heads is 1/2 and the chance of a six is 1/6, so the chance of heads and a six together is 1/12. You can check this by listing all twelve equally likely pairs of outcomes; exactly one of them is heads with a six.

When events are dependent, the multiplication rule needs a conditional probability: P(A∩B) = P(A) × P(B | A), where P(B | A) is the probability of B given that A has already happened. Drawing two aces in a row from a 52-card deck without replacement has probability 4/52 × 3/51 = 1/221, because only three aces remain among 51 cards after the first one is drawn. If the first card is put back and the deck reshuffled, the draws become independent and the probability rises to 4/52 × 4/52 = 1/169.

Events that cannot happen together, such as rolling a 2 and a 5 on the same single die, are called mutually exclusive or disjoint. Their intersection is empty, so P(A∩B) = 0. Do not confuse this with independence: two events with positive probabilities that are mutually exclusive are strongly dependent, because knowing one occurred tells you for certain the other did not.

Union of A and B

The union of A and B, written A∪B, is the event that at least one of them happens: A, B or both. In a Venn diagram it is the whole area covered by either circle. Adding P(A) and P(B) counts the overlap twice, so the general addition rule subtracts it once: P(A∪B) = P(A) + P(B) − P(A∩B).

For independent events this becomes P(A) + P(B) − P(A) × P(B). With 0.5 and 0.4, the union is 0.5 + 0.4 − 0.2 = 0.7. Another way to reach the same answer is through the complement: the only way the union fails is if neither event happens, which has probability 0.5 × 0.6 = 0.3, and 1 − 0.3 = 0.7.

For mutually exclusive events the overlap is zero and the rule reduces to simple addition. The probability of drawing a heart or a spade from a deck is 13/52 + 13/52 = 1/2. The probability of drawing a heart or a king is not 13/52 + 4/52, because the king of hearts belongs to both groups; it is 13/52 + 4/52 − 1/52 = 16/52.

The probability that neither event occurs, P((A∪B)'), is the complement of the union, which for independent events equals P(A') × P(B').

Exclusive OR of A and B

The exclusive or of A and B, written AΔB and called the symmetric difference, is the event that exactly one of them happens: A without B, or B without A, but not both. In a Venn diagram it is the two crescent-shaped regions outside the overlap.

Its probability is the union minus the intersection: P(AΔB) = P(A) + P(B) − 2P(A∩B). The overlap is subtracted twice, once to remove the double count and once more to remove the both case entirely. For independent events with probabilities 0.5 and 0.4, that is 0.5 + 0.4 − 0.4 = 0.5, which also equals 0.5 × 0.6 + 0.5 × 0.4, the chance of A only plus the chance of B only.

Normal Distribution

The normal distribution, also called the Gaussian distribution or bell curve, is the most important continuous probability distribution in statistics. Its curve is symmetric around the mean µ, which is also its median and mode, and its spread is set by the standard deviation σ. Heights of adults, measurement errors and many test scores are approximately normal, and the central limit theorem explains why: the sum or average of many small independent effects tends toward a normal shape, whatever the shape of the individual effects.

For a continuous distribution, the probability of any single exact value is zero. Probabilities belong to ranges, and they equal the area under the density curve between the bounds. The total area under the curve is 1. There is no simple closed formula for these areas, so they are computed numerically from the cumulative distribution function, often written Φ for the standard normal distribution with mean 0 and standard deviation 1. Any normal value x can be converted to the standard scale with the z-score z = (x − µ) ÷ σ.

The empirical rule summarizes the spread. About 68.27% of values lie within one standard deviation of the mean, about 95.45% within two, and about 99.73% within three. That is why values more than three standard deviations from the mean are often treated as outliers. The default inputs, a standard normal distribution between −1 and 1, return 0.68269, the first number of the rule.

As an example, suppose adult male height in a population is roughly normal with a mean of 175 cm and a standard deviation of 7 cm. The probability that a randomly chosen man is between 168 and 182 cm is the area within one standard deviation, about 68%. The probability that he is taller than 189 cm, two standard deviations above the mean, is about 2.3%.

The table produced by the calculator works in the opposite direction: it shows the symmetric range around the mean that contains a given share of the distribution. For 95%, the range is the mean plus or minus 1.96 standard deviations, the familiar multiplier used in confidence intervals and margins of error.

f(x) = (1 ÷ (σ√(2π))) · e^(−(x − µ)² ÷ (2σ²))

µ

the mean, where the curve peaks

σ

the standard deviation, which sets the width

The probability of a range is the area under this curve between the two bounds.

Frequently asked questions