Calculate probability for single events, multiple events (AND/OR), conditional probability, and binomial distribution. Shows fraction, decimal, percentage, and odds.
Probability is the mathematical framework for quantifying uncertainty — expressing how likely events are to occur on a scale from 0 (impossible) to 1 (certain). Our free probability calculator handles single events, multiple events (AND/OR), conditional probability, Bayes' theorem, and the binomial distribution, making it ideal for students in the UK studying GCSE and A-Level Statistics, US students in AP Statistics, and anyone needing quick probability calculations.
For equally likely outcomes, the probability of an event is:
P(event) = Number of favourable outcomes / Total number of outcomes
Examples:
The complement rule: P(not A) = 1 − P(A). The probability of NOT rolling a 3 is 1 − 1/6 = 5/6.
If two events A and B are independent (the outcome of one does not affect the other):
P(A and B) = P(A) × P(B)
Example: Rolling a 3 on a die AND flipping heads: P = (1/6) × (1/2) = 1/12 ≈ 0.083
If two events are mutually exclusive (cannot both happen at the same time):
P(A or B) = P(A) + P(B)
Example: Rolling a 2 OR a 5 on a fair die: P = 1/6 + 1/6 = 2/6 = 1/3
When events can overlap (both can occur simultaneously):
P(A or B) = P(A) + P(B) − P(A and B)
For independent events: P(A and B) = P(A) × P(B), so: P(A or B) = P(A) + P(B) − P(A)×P(B)
Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred:
P(A|B) = P(A∩B) / P(B)
Example: In a class of 30 students, 18 passed maths and 12 passed both maths and science. Given a student passed maths, what is the probability they also passed science? P(science|maths) = 12/18 = 2/3 ≈ 0.667.
Bayes' theorem allows you to update probabilities when new evidence becomes available:
P(A|B) = P(B|A) × P(A) / P(B)
A classic example is medical testing. If a disease affects 1% of a population, a test has 95% sensitivity (true positive rate) and 5% false positive rate, and a random person tests positive, what is the probability they actually have the disease?
P(disease) = 0.01, P(positive|disease) = 0.95, P(positive|no disease) = 0.05
P(positive) = 0.95×0.01 + 0.05×0.99 = 0.0095 + 0.0495 = 0.059
P(disease|positive) = (0.95 × 0.01) / 0.059 ≈ 16.1%
This counter-intuitive result — only 16% chance of actually having the disease despite a positive test — demonstrates why base rates matter enormously in medical screening.
The binomial distribution models the number of successes in n independent trials, each with probability p of success:
P(X = k) = C(n,k) × p^k × (1−p)^(n−k)
Where C(n,k) = n! / (k! × (n−k)!) is the binomial coefficient.
Example: If a student randomly guesses on a 10-question multiple-choice test (4 options each, p = 0.25), what is the probability of getting exactly 3 correct?
P(X=3) = C(10,3) × (0.25)³ × (0.75)⁷ = 120 × 0.015625 × 0.133484 ≈ 0.2503 (25%)
The binomial distribution has mean = np and variance = np(1−p). For n=10, p=0.25: mean = 2.5 correct answers, SD = √(10×0.25×0.75) ≈ 1.37.
Probability and odds are related but different ways of expressing likelihood:
| Measure | Formula | Example (P=0.25) |
|---|---|---|
| Probability | P = favourable / total | 0.25 (25%) |
| Odds for (in favour) | P / (1−P) | 0.25/0.75 = 1:3 ("one to three") |
| Odds against | (1−P) / P | 0.75/0.25 = 3:1 ("three to one against") |
In UK gambling, odds are typically expressed as fractional odds (e.g., 3/1 means win £3 for every £1 staked). In the US, moneyline odds are used (+300 means win $300 on a $100 bet). Both formats can be converted to implied probability.
UK National Lottery (Lotto): Choose 6 from 59. Odds of jackpot = C(59,6) = 1 in 45,057,474 ≈ 0.0000022%.
US Powerball: Choose 5 from 69 plus 1 from 26. Odds = C(69,5) × 26 = 292,201,338. Approximately 1 in 292 million ≈ 0.00000034%.
US Mega Millions: Choose 5 from 70 plus 1 from 25. Odds = 1 in 302,575,350.
At GCSE Mathematics, students cover basic probability, mutually exclusive events, independent events, two-way tables, tree diagrams, and Venn diagrams. A-Level Mathematics and Statistics cover conditional probability, Bayes' theorem (in some specifications), the binomial distribution, normal distribution approximations, and hypothesis testing using probability. AQA, Edexcel, and OCR all include probability prominently in both GCSE and A-Level syllabuses.
AP Statistics covers probability in Units 4–6, including probability rules, conditional probability, independent events, combining random variables, and probability distributions including binomial and geometric. The AP exam typically includes 4–6 free-response questions and 40 multiple-choice questions, with probability appearing throughout. Understanding when to apply multiplication vs addition rules is a key exam skill.
P(event) = Number of favourable outcomes / Total number of outcomes. This assumes all outcomes are equally likely. For a fair coin: P(heads) = 1/2. For a fair die: P(rolling 4) = 1/6.
AND (intersection) uses the multiplication rule: P(A and B) = P(A)×P(B) for independent events. OR (union) uses the addition rule: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events P(A and B) = 0, so P(A or B) = P(A) + P(B).
P(A|B) is the probability of A given B has occurred: P(A|B) = P(A∩B) / P(B). Knowing B has occurred changes the probability of A if A and B are not independent.
The binomial distribution models the number of successes in n independent trials, each with probability p of success. It applies when: you have a fixed number of trials; each trial is independent; each trial has only two outcomes (success/failure); the probability of success is constant.
Odds for = P / (1−P). Odds against = (1−P) / P. For P = 0.25: odds for are 1:3 (1 to 3), odds against are 3:1. Bookmakers use fractional odds (UK) or moneyline odds (US), both convertible to implied probability.
Two events are independent if the occurrence of one does not affect the probability of the other. Formally: P(A|B) = P(A), equivalently P(A∩B) = P(A)×P(B). Rolling two dice: the result of the first die does not affect the second. Drawing with replacement from a deck: independent. Drawing without replacement: not independent (dependent events).
Bayes' theorem updates the probability of a hypothesis given new evidence: P(hypothesis|evidence) = P(evidence|hypothesis) × P(hypothesis) / P(evidence). It combines your prior belief about the hypothesis with the likelihood of the evidence under that hypothesis to give an updated (posterior) probability.
Yes. GCSE Mathematics (all UK exam boards) covers basic probability, mutually exclusive and independent events, Venn diagrams, tree diagrams, and conditional probability through two-way tables. A-Level Mathematics extends this to the binomial distribution, hypothesis testing, and formal conditional probability.
Results are for educational purposes. Probability calculations assume stated conditions hold exactly. Real-world outcomes involve additional factors not captured by simple models.