Probability Calculator

Calculate probabilities for any scenario. Choose the type of probability you need from the tabs below: simple events, combined events with AND and OR, conditional probability, Bayes' theorem, permutations and combinations, binomial distribution, normal distribution (z-scores), and Poisson distribution. Each calculator includes the formula and a worked example in plain English.

Basic Probability
P(A) = number of favourable outcomes / total possible outcomes
P(A) = favourable / total
Example
A standard six-sided die is rolled. What is the probability of rolling a 4? Favourable outcomes = 1 (only the number 4). Total outcomes = 6. P(4) = 1/6 = 0.1667 = 16.67%.
Combined Events: AND and OR
Calculate the probability of two events occurring together or at least one occurring
Example
A bag has 10 balls: 4 red and 3 blue. Two balls are drawn one at a time with replacement. P(red) = 0.4, P(blue) = 0.3. Because the draws are independent: P(red AND blue) = 0.4 x 0.3 = 0.12. P(red OR blue) = 0.4 + 0.3 - 0.12 = 0.58.
Conditional Probability
The probability of A given that B has already occurred: P(A|B) = P(A and B) / P(B)
P(A|B) = P(A and B) / P(B)
Example
In a class of 30 students, 12 play sport and 8 play both sport and music. What is the probability that a student plays music, given they play sport? P(sport and music) = 8/30 = 0.267. P(sport) = 12/30 = 0.4. P(music | sport) = 0.267 / 0.4 = 0.667 = 66.7%.
Bayes' Theorem
Update the probability of a hypothesis given new evidence
P(A|B) = P(B|A) x P(A) / [ P(B|A) x P(A) + P(B|A') x P(A') ]
Medical test example
A disease affects 1% of the population. A test for the disease is 99% accurate (true positive rate). However, 5% of healthy people also test positive (false positive rate). If a person tests positive, what is the probability they actually have the disease? Enter P(A)=0.01, P(B|A)=0.99, P(B|A')=0.05. Result: despite the positive test, there is only about a 16.7% chance the person actually has the disease. This surprising result, known as the base rate fallacy, is why doctors do not rely on a single test for rare diseases.
Permutations and Combinations
Count ordered arrangements (permutations) or unordered selections (combinations)
Examples
Permutations (order matters): How many ways can 3 runners finish in 1st, 2nd, and 3rd from a field of 10? P(10,3) = 10!/7! = 10 x 9 x 8 = 720 arrangements.

Combinations (order does not matter): How many different groups of 3 can be chosen from 10 people? C(10,3) = 10!/(3! x 7!) = 120 different groups. The NZ Lotto 6/40 has C(40,6) = 3,838,380 possible combinations.
Binomial Distribution
P(X = k) = C(n,k) x p^k x (1-p)^(n-k) | Fixed trials, two outcomes, constant probability
P(X = k) = C(n,k) x p^k x (1-p)^(n-k)
Example
A fair coin is flipped 10 times. What is the probability of getting exactly 6 heads? n=10, p=0.5, k=6. P(X=6) = C(10,6) x 0.5^6 x 0.5^4 = 210 x 0.015625 x 0.0625 = 0.2051 = 20.5%. Useful for: quality control, sports statistics, drug trial analysis, genetics.
Normal Distribution
The bell curve. Most natural measurements follow a normal distribution.
z = (x - mean) / standard_deviation P = normalCDF(z)
Example
Heights of NZ adult males are approximately normally distributed with mean 178cm and standard deviation 7cm. What proportion of men are taller than 190cm? Set mean=178, SD=7, x=190, find P(X >= 190). Z = (190-178)/7 = 1.714. P(X >= 190) = 1 - normalCDF(1.714) = 4.3%. About 4.3% of NZ adult men are taller than 190cm.
Poisson Distribution
P(X = k) = e^(-lambda) x lambda^k / k! | Events occurring at a constant average rate
P(X = k) = e^(-lambda) x lambda^k / k!
Example
A call centre receives an average of 3 calls per hour. What is the probability of receiving exactly 2 calls in a given hour? Lambda=3, k=2. P(X=2) = e^(-3) x 3^2 / 2! = 0.0498 x 9 / 2 = 0.2240 = 22.4%. Poisson is used for: call centres, hospital admissions, website traffic spikes, number of accidents on a road per month.

What is probability?

Probability is a number between 0 and 1 (or 0% and 100%) 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. A probability of 0.5 means the event occurs half the time. Probabilities are always calculated as: number of ways the event can happen, divided by the total number of equally likely outcomes.

Types of probability

Classical probability applies when all outcomes are equally likely, such as rolling a fair die or drawing a card from a shuffled deck. Empirical probability is based on observed data: if a factory produced 12 defective items out of 1,000, the empirical probability of a defect is 1.2%. Subjective probability is an educated estimate based on judgment, used in forecasting and risk assessment.

The addition rule (OR)

P(A or B) = P(A) + P(B) - P(A and B). The reason we subtract P(A and B) is to avoid counting the overlap twice. If events are mutually exclusive (they cannot both happen at the same time), then P(A and B) = 0 and the formula simplifies to P(A or B) = P(A) + P(B).

The multiplication rule (AND)

For two independent events (where one does not affect the other): P(A and B) = P(A) x P(B). For dependent events: P(A and B) = P(A) x P(B|A), where P(B|A) is the conditional probability of B given that A has occurred. Independence is a crucial assumption: rolling two dice is independent (first roll does not affect second), but drawing cards without replacement is dependent (removing a card changes the remaining pool).

The normal distribution and the 68-95-99.7 rule

The normal distribution (bell curve) is the most important probability distribution in statistics because many real-world measurements naturally follow it: heights, exam scores, measurement errors, financial returns. The 68-95-99.7 rule states: approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. If exam scores have a mean of 65 and standard deviation of 10, then 95% of students score between 45 and 85 (mean plus or minus two standard deviations).

Bayes' theorem and the base rate fallacy

Bayes' theorem is the mathematically correct way to update a probability when new evidence arrives. The base rate fallacy is the common mistake of ignoring how rare an event is when interpreting a positive test result. A medical test that is 99% accurate sounds very reliable, but if the disease it tests for affects only 1% of the population, then a positive result only means about a 17% chance of actually having the disease. The high false positive rate among healthy people (who are 99% of the population) overwhelms the accurate positives. Bayes' theorem is used in medical diagnosis, spam email filtering, financial risk models, and machine learning classification.

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