Probability Calculator

Probability quantifies how likely an event is to occur. It ranges from 0 (impossible) to 1 (certain).

In everyday terms, a 0.5 probability means a 50-50 chance.

Our calculator handles all standard probability types used in statistics, math, gambling analysis, and science.

• Basic probability divides favorable outcomes by total outcomes.
• For multiple independent events, multiply probabilities (AND) or use the union formula (OR).
• Conditional probability asks 'what if we already know B happened?'
• Binomial probability handles repeated independent trials like flipping a coin n times.

To calculate the probability of a single event, divide the number of favorable outcomes by the total number of equally likely outcomes: P(A) = favorable / total.

For example, drawing a red card from a standard 52-card deck gives P = 26/52 = 1/2 = 50%. This classical definition assumes every outcome is equally likely, an idea formalized by Pierre-Simon Laplace and described by Encyclopaedia Britannica as the foundation of probability theory.

The result always lies between 0 and 1, or 0% and 100%. When outcomes are not equally likely, probability is instead estimated from long-run relative frequencies observed across many repeated trials.

Three rules cover most probability problems.

• The complement rule states P(not A) = 1 - P(A).
• The multiplication rule for independent events gives P(A and B) = P(A) x P(B); for dependent events it uses conditional probability, P(A and B) = P(A) x P(B|A).
• The general addition rule is P(A or B) = P(A) + P(B) - P(A and B), which subtracts the overlap counted twice.

As Wolfram MathWorld notes, when events are mutually exclusive the intersection is zero, so the formula simplifies to P(A) + P(B). These rules combine to solve compound scenarios such as multi-stage draws or repeated trials.

Conditional probability, written P(A|B), is the probability of A occurring given that B has already happened: P(A|B) = P(A and B) / P(B).

Bayes' theorem reverses this relationship, expressing P(A|B) = [P(B|A) x P(A)] / P(B). It lets you update a prior belief with new evidence, and Khan Academy highlights it as the engine behind medical test interpretation and spam filtering.

A classic result is base rate neglect: even a 99% accurate test for a rare disease affecting 1 in 1,000 people yields mostly false positives, because the small prior probability dominates. Bayesian reasoning corrects this by weighting evidence against how common the condition truly is.

The binomial formula gives the probability of exactly k successes in n independent trials, each with success probability p: P(X=k) = C(n,k) x p^k x (1-p)^(n-k), where C(n,k) is the number of combinations.

For example, the chance of exactly 3 heads in 5 fair coin flips is C(5,3) x 0.5^3 x 0.5^2 = 10 x 0.125 x 0.25 = 0.3125, or 31.25%. The NIST/SEMATECH e-Handbook of Statistical Methods documents the binomial distribution's mean, np, and variance, np(1-p).

The binomial model applies whenever:

• trials are independent,
• the count is fixed,
• and each trial has only two outcomes, success or failure.

For a single fair die, each face has probability 1/6. With two dice there are 36 equally likely ordered outcomes, so the most common sum, 7, occurs 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), giving P(sum=7) = 6/36 = 1/6.

A standard 52-card deck has 4 suits and 13 ranks: P(any Ace) = 4/52 = 1/13, and P(a Heart) = 13/52 = 1/4.

Multi-step draws use the multiplication rule; for instance, two Aces without replacement is 4/52 x 3/51 = 12/2652, about 0.45%. Britannica traces these counting methods to the correspondence between Fermat and Pascal on games of chance.

Two events are independent when one occurring does not change the probability of the other; formally, P(A|B) = P(A). Flipping a coin twice is independent because the first flip has no memory of the second.

Events are dependent when the outcome of one alters the sample space for the next, which is exactly what happens in sampling without replacement. Wolfram MathWorld emphasizes that independence must be verified, not assumed.

A frequent classroom mistake is multiplying probabilities as if events were independent when they are actually linked, such as drawing cards without returning them, which inflates or deflates the true chance of a compound outcome.

Probability drives decisions across many fields.

• Insurers price policies using the expected frequency and severity of claims.
• Medical researchers report treatment effects and diagnostic accuracy as conditional probabilities.
• Meteorologists express a 30% chance of rain as the fraction of similar past conditions that produced rain.

In finance, probability underlies risk models and option pricing, while quality-control engineers use binomial and Poisson models to set defect tolerances. Machine learning classifiers, as covered by Khan Academy and modern statistics courses, output probabilities for each label.

Even everyday choices, from carrying an umbrella to buying a warranty, are implicit probability calculations weighing likelihood against cost.

The most common errors stem from misjudging independence and prior odds.

• The gambler's fallacy assumes past results affect independent future trials, so people expect heads after a run of tails, though each flip stays 50%.
• Base rate neglect ignores how rare an event is when interpreting evidence, distorting test-result reasoning.
• Confusing P(A|B) with P(B|A), the inverse-probability error, reverses cause and effect.
• Others forget to subtract the overlap in the OR rule, double-counting shared outcomes, or add probabilities of events that are not mutually exclusive.

The counterintuitive birthday problem, in which just 23 people give a 50.7% chance of a shared birthday, shows how intuition often understates compound probabilities.

Expected value is the long-run average outcome of a random process, computed by multiplying each outcome by its probability and summing the results: E(X) = Sum of x_i x P(x_i).

For a single fair die the expected value is (1+2+3+4+5+6)/6 = 3.5, even though 3.5 can never actually be rolled. Wolfram MathWorld defines it as the first moment of a distribution, and it is central to decision-making under uncertainty.

A game or bet with negative expected value loses money on average over time, which is why casino games are engineered with a house edge. Expected value connects raw probabilities to practical, quantifiable decisions.