Probability Calculator
Calculate single event probability, complementary probability, and odds for/against an event.
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How to use this calculator
Probability ranges from 0 (impossible) to 1 (certain). Expressed as fraction, decimal, or percentage.
- 1
Enter the number of favourable outcomes (the event you are interested in).
- 2
Enter the total number of possible outcomes.
- 3
Optionally enter the number of trials for the "at least one success" calculation.
Frequently asked questions
What does “favourable outcomes” mean here?
It is the count of outcomes that make your event happen, out of the total number of equally likely outcomes. For one card drawn from a standard 52-card deck, the event “draw an ace” has 4 favourable outcomes out of 52, which the calculator reduces to 1/13 (about 7.6923%). The method only holds when every outcome really is equally likely.
Why is my fraction different from what I typed?
Whole-number inputs are reduced to lowest terms, so 6 favourable out of 36 total is shown as 1/6, and 3 out of 6 as 1/2. The percentage and decimal are unaffected. If either number has a decimal part, such as 2.5 favourable, the fraction is shown exactly as entered and the odds are not reduced.
How are the odds written, and what happens at 0 and 100%?
Odds for are favourable : unfavourable, reduced when both are whole numbers, so 1 favourable out of 6 gives 1 : 5 for and 5 : 1 against. With 0 favourable the odds for are 0 : 1 and against are 1 : 0. When favourable equals total, odds for are 1 : 0, meaning the event cannot fail.
How is the “at least one success” row calculated?
It uses 1 − (1 − p)ⁿ, where n is the number of trials, and treats each trial as independent with the same probability. Rolling a six on one die in 4 rolls gives 51.77%; drawing an ace from a 52-card deck in 10 draws, replacing the card each time, gives 55.09%. It is shown to two decimal places.
Can I use it for draws without replacement?
Not for the trials row. That row assumes the probability stays the same on every trial, which is true for dice, coin flips and draws with replacement. Without replacement the probability changes after each draw, so the figure would be wrong. The single-event probability at the top is unaffected.
What input is rejected?
Total outcomes of 0 or less, negative favourable outcomes, and favourable outcomes greater than the total each return an error. If the trials box is zero (which is also what a cleared box becomes), negative or not a whole number, the main results still appear and the trials row asks for a whole number of 1 or more.
Does it combine two events, such as A and B or A or B?
No. It handles one event defined by a favourable count and a total count. For two independent events you would multiply their probabilities for “both”, but that step is not performed here; run each event separately and combine the decimals yourself.
How are results rounded?
The probability and complement show up to four decimals as percentages, the decimal form shows up to eight places, and the at-least-once row shows two. Trailing zeros are dropped, so 50.0000% appears as 50%.
Single-event probability, odds and repeated trials
From counts to probability, odds and complement
Give the calculator a favourable count and a total count and it returns the probability as a percentage, a decimal and a reduced fraction. It also shows the complement, which is 1 minus the probability, and the odds written both ways. Probability compares favourable outcomes with all outcomes, while odds compare favourable with unfavourable, so a probability of 1/6 is odds of 1 : 5, not 1 : 6. That distinction is the most common source of mistakes when reading betting-style odds.
| Event | Favourable / total | Probability | Odds for | Complement |
|---|---|---|---|---|
| Ace from a 52-card deck | 4 / 52 | 7.6923% | 1 : 12 | 92.3077% |
| Six on one die | 1 / 6 | 16.6667% | 1 : 5 | 83.3333% |
| Sum of 7 on two dice | 6 / 36 | 16.6667% | 1 : 5 | 83.3333% |
| Impossible event | 0 / 5 | 0% | 0 : 1 | 100% |
Worked example: at least one success
The trials field answers “what are the chances it happens at least once?” Take the six on a die. One roll gives 16.6667%. Over 4 rolls the calculator reports 51.77%, which is just over even, and the cause is that the chance of missing four times in a row is (5/6)⁴, about 48.2%, so the remainder is the success side. Set the die example to 1 trial and the figure collapses back to 16.67%, a quick check that the row is consistent.
Worked example: drawing an ace repeatedly
Enter 4 favourable and 52 total with 10 trials. The single draw is 7.6923%, reduced to 1/13 with odds of 1 : 12 for. Across 10 draws where the card is put back and the deck reshuffled, the chance of seeing at least one ace is 55.09%. Notice that ten tries do not bring you near certainty; probabilities of repeated events grow more slowly than most people expect, and they never reach 100% unless a single trial is already certain.
Where equal likelihood breaks down
The favourable-over-total rule needs equally likely outcomes. “It will either rain or not” has two outcomes, but they are not equally likely, so entering 1 and 2 would give a meaningless 50%. Weighted dice, biased coins and real-world events need probabilities estimated from data rather than counted. Likewise, the repeated-trials row requires independence: each trial must not affect the next.
Reading the output correctly
A probability of 0% means the event cannot happen under your counts, and 100% means it always does. Everything in between should be read alongside the trial count: a 5% event is rare on one attempt, yet likely to appear at least once over enough attempts. The complement row is useful for the opposite question, such as the chance a batch of items contains no defects at all.
Learn more from an authoritative source:
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Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →