Probability Calculator - Single, Two Events & Repeated

About the Probability Calculator

Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain), often written as a percentage. It underpins games of chance, weather forecasts, insurance, medical testing and every kind of statistical analysis. Yet simple-looking questions — what is the chance of rolling at least one six in four throws? — trip up many people, because probabilities do not simply add up.

This probability calculator handles the three most common situations. For a single event with equally likely outcomes, it finds the probability, its complement, the odds and how often to expect it. For two independent events, it finds the chance of both, either, exactly one and neither. For repeated tries, it finds the chance of at least one success, of never succeeding and of succeeding every time, plus the number of tries needed for a better-than-even chance.

How to Use the Probability Calculator

Choose Single event, Two events or Repeated tries.

For a single event, enter the number of favourable outcomes and the total number of equally likely outcomes — for example 1 and 6 for rolling a six.

For two events, enter the probability of A and the probability of B as percentages.

For repeated tries, enter the chance per try and the number of tries.

The Formulas

  single event        P(A) = favourable outcomes ÷ total outcomes
  complement          P(not A) = 1 − P(A)
  both (independent)  P(A and B) = P(A) × P(B)
  either              P(A or B) = P(A) + P(B) − P(A and B)
  at least once       P = 1 − (1 − p)^n

Step-by-Step Example: A Single Event

The chance of drawing an ace from a shuffled deck.

  Favourable: 4 aces    Total: 52 cards
  P = 4 ÷ 52 = 1/13 = 0.0769 → 7.69%
  Not an ace: 92.31%    Odds: 12 to 1 against

On average you would draw an ace once in every 13 draws (replacing and shuffling each time).

Step-by-Step Example: Two Events

A has a 50% chance and B a 30% chance, independently.

  Both:         0.5 × 0.3 = 0.15               → 15%
  Either:       0.5 + 0.3 − 0.15 = 0.65        → 65%
  Exactly one:  0.65 − 0.15 = 0.5              → 50%
  Neither:      1 − 0.65 = 0.35                → 35%

The four answers are consistent: both (15%) + exactly one (50%) + neither (35%) = 100%.

Step-by-Step Example: Repeated Tries

A 10% chance per try, over 10 tries.

  Never:           0.9^10 = 0.3487   → 34.87%
  At least once:   1 − 0.3487        → 65.13%
  Expected:        10 × 0.1 = 1 success on average

Many people guess 100 percent, by adding 10 percent ten times. The real answer is 65.13% — about a third of the time, ten tries bring no success at all. It takes 7 tries before the chance of at least one success passes 50 percent.

Independent and Dependent Events

The two-event formulas assume independence: knowing whether A happened tells you nothing about B. Coin tosses, dice rolls and draws with replacement are independent. Drawing cards without replacement is not — after one ace is drawn, only 3 remain among 51 cards, so the chance of a second ace drops to 3/51. Rain today and rain tomorrow are not independent either. For dependent events, multiply by the conditional probability: P(A and B) = P(A) × P(B given A).

Mutually Exclusive Events

Events that cannot happen together, such as rolling a 2 and rolling a 5 on the same throw, are mutually exclusive. For them, P(A or B) is simply P(A) + P(B), because there is no overlap to subtract: 1/6 + 1/6 = 1/3. Do not confuse this with independence — mutually exclusive events are strongly dependent, because one happening rules out the other.

The Complement Trick

Questions about "at least one" are almost always easiest through the complement: work out the chance of none, and subtract from one. The famous birthday problem uses the same idea — in a group of 23 people, the chance that no two share a birthday is just under 50 percent, so the chance that at least two do is just over half, far higher than most people expect.

Small Risks Add Up

The repeated-tries formula explains why small risks matter over time. A 1 percent chance of something going wrong on any given day sounds negligible, but over a year of 365 independent days the chance of it happening at least once is 1 − 0.99^365, about 97.45 percent. Engineers use the same reasoning for component failures, and insurers for rare events such as floods: a "1 in 100 year" flood has about a 26 percent chance of occurring at least once during a 30-year mortgage.

Probability, Percentages and Odds

The same chance can be written in several ways. A probability of 0.25 is 25 percent, 1 in 4, or odds of 3 to 1 against. Weather forecasts use percentages, bookmakers use odds, and scientists use decimals between 0 and 1. The calculator shows the main forms side by side so you can move between them easily.

Understanding Your Result

For a single event, the headline is the probability; the fraction, complement, odds and how often lines express it in other ways.

For two events, the headline is the chance of both; the either, exactly one and neither lines complete the picture.

For repeated tries, the headline is the chance of at least one success; the never, every time, expected count and even chance lines add detail.

When Should You Use This Calculator?

Use it for dice, cards, coins and other games of chance.

Use it for homework on basic, combined and repeated probability.

Use it to judge risks that repeat, such as a small daily chance over a year.

Use it to check intuition before making a decision under uncertainty.

Common Mistakes

Adding probabilities for repeated tries. Use 1 − (1 − p)^n instead.

Multiplying dependent probabilities as if independent. Use conditional probabilities.

Forgetting to subtract the overlap in P(A or B). Otherwise both-at-once cases are counted twice.

Assuming outcomes are equally likely when they are not. The favourable-over-total rule needs equally likely outcomes.

The gambler's fallacy. Past independent results do not change the next one; a coin that has landed heads five times is still 50–50.