Probability Calculator
Last updated: 27 June 2026
Reviewed by Gavin Meiring, Lead research and primary author · Doctoral Candidate (Corporate Governance) · Research and drafting assisted by AI
- The probability of winning the UK National Lottery jackpot is about 1 in 45 million. You are more likely to be struck by lightning twice in your lifetime.
- The 'birthday problem': in a room of just 23 people, there's a 50% chance two share a birthday. Most people guess 183. Our intuition about probability is consistently wrong.
- In poker, a royal flush has a probability of 1 in 649,740. Most professional poker players never see one during their career — yet they're depicted constantly in films.
Probability Calculator
A probability calculator computes the likelihood of events occurring, including single events, combined events, and conditional probabilities. It is used by students, data scientists, statisticians, and anyone needing to quantify uncertainty or analyse chance.
How to Use the Probability Calculator
- Select the type of probability calculation: single event, combined events (AND/OR), or conditional probability.
- Enter the number of favourable outcomes and the total possible outcomes for a single event.
- For combined events, enter the probabilities of both events and specify whether they are independent or mutually exclusive.
- Click Calculate to see the probability as a fraction, decimal, and percentage.
- Review the explanation panel which describes the logic used in plain language.
The Formula
Probability of a single event: P(A) = number of favourable outcomes divided by total possible outcomes. The result ranges from 0 (impossible) to 1 (certain). For two independent events A and B: P(A AND B) = P(A) multiplied by P(B). For mutually exclusive events: P(A OR B) = P(A) plus P(B). For non-mutually exclusive events: P(A OR B) = P(A) plus P(B) minus P(A AND B). Conditional probability: P(A given B) = P(A AND B) divided by P(B).
Real-World Example
A bag contains 5 red balls, 3 blue balls, and 2 green balls (10 total). What is the probability of drawing a red ball, then a blue ball without replacement? P(red first) = 5 divided by 10 = 0.5. After removing one red ball, 9 remain with 3 blue. P(blue second given red first) = 3 divided by 9 = 0.333. P(red then blue) = 0.5 multiplied by 0.333 = 0.167 or approximately 16.7%. This is a conditional probability where the events are dependent.
Probability in Everyday Decision Making
Probability underpins risk assessment, weather forecasting, insurance pricing, medical diagnostics, and financial modelling. A weather forecast of 70% rain means that in similar atmospheric conditions, it has rained 70% of the time historically. Medical test sensitivity and specificity are probabilities that determine how reliable a diagnosis is. Understanding the difference between probability and odds (odds = probability divided by (1 minus probability)) is important when interpreting sporting odds or betting markets.
Frequently Asked Questions
What is the difference between theoretical and experimental probability? Theoretical probability is calculated from known outcomes (flipping a fair coin has 0.5 probability of heads). Experimental probability is calculated from actual observed results; if you flip a coin 100 times and get 47 heads, the experimental probability is 0.47. With enough trials, experimental probability converges to theoretical probability.
What does mutually exclusive mean in probability? Two events are mutually exclusive if they cannot both occur at the same time. Rolling a 3 and rolling a 5 on a single die roll are mutually exclusive. Being in London and being in Paris simultaneously (for the same person) is another example. Mutually exclusive events satisfy P(A AND B) = 0.
What is the complement rule? The complement rule states that the probability of an event not occurring is 1 minus the probability of it occurring: P(not A) = 1 minus P(A). This is useful when the complement is easier to calculate. For example, the probability of rolling at least one 6 in 4 dice rolls = 1 minus (5/6 to the power of 4) = 1 minus 0.482 = 0.518.
What is a probability distribution? A probability distribution describes the probability of all possible outcomes of a random variable. Discrete distributions (like the binomial distribution) apply to countable outcomes. Continuous distributions (like the normal distribution) describe outcomes that can take any value within a range. Distributions are fundamental to statistics and inference.
When do I add probabilities instead of multiplying them?
Add when the question asks for one event or the other, and multiply when it asks for both. Addition needs mutually exclusive events, or the overlap subtracted once. Multiplication needs independent events, or the conditional probability of the second given the first. Plug your own figures into the calculator above.
What does a probability of 0 or 1 actually mean?
Zero means the event cannot happen under the assumptions you set, and one means it must. Both are statements about the model rather than about the world. Assigning zero to an outcome rules it out completely, which is a strong claim to rest on data alone.
Why does the same event have a different probability in two studies?
Sample size, measurement, and the definition of the event all move the answer. Two studies of the same rare outcome can differ several fold without either being wrong. Check the sample size and the stated definition before comparing the numbers, and treat a single study as one estimate rather than the estimate.
Worked example: two dice and the complement rule
Two fair dice give 36 equally likely pairs. The chance of a six on both is 1/36, because the two rolls are independent and the probabilities multiply.
The chance of at least one six in four rolls is easier to reach from the other direction. The chance a single roll is not a six is 5/6. Four independent rolls with no six at all is (5/6)^4 = 0.4823, and subtracting that from 1 gives 0.5177, or 51.77%. The complement rule turns four cases into one multiplication.
| Question | Working | Result |
|---|---|---|
| A six on both of two dice | (1/6) x (1/6) | 0.0278, or 2.78% |
| No six in four rolls | (5/6)^4 | 0.4823, or 48.23% |
| At least one six in four rolls | 1 - (5/6)^4 | 0.5177, or 51.77% |
| A 3 or a 5 on one die | 1/6 + 1/6 | 0.3333, or 33.33% |
| A heart or a face card | 13/52 + 12/52 - 3/52 | 0.4231, or 42.31% |
The last row is the one people get wrong. Hearts and face cards overlap on the jack, queen, and king of hearts, so the two counts share three cards and the shared part is subtracted once. Adding the two counts without the subtraction gives 0.4808 and overstates the answer.
Conditional probability with a real test
A medical screening example shows why the order of the information matters. A disease affects 1% of a population. A test detects it in 99% of people who have it and returns a false positive in 5% of people who do not.
| Outcome | Working | Probability |
|---|---|---|
| Has the disease and tests positive | 0.99 x 0.01 | 0.0099 |
| Does not have it and tests positive | 0.05 x 0.99 | 0.0495 |
| Tests positive in total | 0.0099 + 0.0495 | 0.0594 |
| Has the disease, given a positive test | 0.0099 / 0.0594 | 0.1667, or 16.67% |
A positive result on a 99% accurate test still leaves a 16.67% chance of having the disease. The false positives outnumber the true ones because the disease is rare. This is the base rate effect, and it explains why a positive screening result is usually followed by a second, more specific test rather than a diagnosis.
Independence and mutual exclusivity are not the same thing
Two events are independent when the occurrence of one leaves the probability of the other unchanged. They are mutually exclusive when they cannot both happen. Mutually exclusive events are never independent unless one of them has probability zero, because knowing that one occurred tells you the other did not.
| Pair of events | Relationship | Rule that applies |
|---|---|---|
| A 3 and a 5 on one die roll | mutually exclusive | P(A or B) = 1/6 + 1/6 |
| A six on die one and a six on die two | independent | P(A and B) = 1/6 x 1/6 |
| Drawing two red cards without replacement | dependent | P(second red) changes after the first draw |
| Rain today and rain tomorrow | neither | needs historical joint frequencies |
A case where the answer surprises people
With 23 people in a room, the chance that two of them share a birthday is 50.73%. The calculation multiplies the chance that each new person avoids every birthday already taken.
| People in the room | Probability of a shared birthday |
|---|---|
| 10 | 11.69% |
| 20 | 41.14% |
| 23 | 50.73% |
| 30 | 70.63% |
| 40 | 89.12% |
| 50 | 97.04% |
The figure assumes 365 equally likely birthdays and ignores leap days, twins, and seasonal birth patterns. Real data shifts the numbers a little, not enough to change the point that probabilities of this kind grow faster than intuition suggests.
Converting between probability and odds
Odds compare the number of ways an event happens with the number of ways it does not. Probability compares the favourable ways with all the ways.
Odds in favour = P / (1 - P) Probability = odds in favour / (1 + odds in favour)
| Probability | Odds in favour | Odds against |
|---|---|---|
| 1/6 = 0.1667 | 1 to 5 | 5 to 1 |
| 0.25 | 1 to 3 | 3 to 1 |
| 0.50 | 1 to 1 | 1 to 1 |
| 0.70 | 7 to 3 | 3 to 7 |
| 0.75 | 3 to 1 | 1 to 3 |
| 5/6 = 0.8333 | 5 to 1 | 1 to 5 |
Quoted betting odds need one more step, because a bookmaker's prices do not add up to a probability distribution. Convert each price to its implied probability first, add them, and compare the total with 1. The excess is the margin, and it is the reason a set of prices cannot be read as probabilities directly.
Sources
- Khan Academy, Probability library. https://www.khanacademy.org/math/statistics-probability/probability-library. Accessed 2026-09-13.
- NIST/SEMATECH e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/. Accessed 2026-09-13. Definitions of probability, independence, and conditional probability.
Also try these free tools related to Probability Calculator: - Standard Deviation Calculator