Statistics
Probability Calculator
Find the probability of a single event, combine two events with AND / OR / NEITHER, or work out the odds over a series of repeated trials — every result live, with the steps and an odds lookup chart.
Count the outcomes
The ways the event can happen
All equally likely possibilities
Probability
0.25= 25%
Unlikely· 1 in 4
0 · impossible0.5 · even1 · certain
Percent
25%
Fraction
1/4
Odds
1 in 4
Try an example
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The basics
What is probability?
Probability measures how likely something is, on a scale from 0 to 1. It is the share of all equally likely outcomes that count as the one you want. A probability of 0 means it can never happen; 1 means it always does; everything in between is a fraction of certainty you can read as a decimal, a percent, a fraction, or odds.
P = favorable / total
The basic ratio
Count the outcomes you want, divide by every equally likely outcome there is. As long as each outcome is just as likely, that ratio is the probability of the event A.
13 hearts / 52 cards = 0.25
0 ≤ P ≤ 1
Always between 0 and 1
Because favorable outcomes are a subset of all outcomes, the ratio can never exceed one or drop below zero. Multiply by 100 to read any probability as a percentage instead.
0 = never · 0.5 = even · 1 = certain
Method
How to calculate probability by hand
Three steps turn a question into a number you can read four different ways.
- 1
Count the favorable outcomes
How many distinct results satisfy the event A you care about? For drawing a heart, that is the 13 hearts in the deck.
favorable = 13 - 2
Count every equally likely outcome
How many results are possible in total, each just as likely as the next? A full deck has 52 cards.
total = 52 - 3
Divide, then convert
Divide favorable by total. Multiply by 100 for a percent, or reduce the ratio for a fraction; invert it for odds.
13 / 52 = 0.25 = 25% = 1/4
Combining events
AND, OR, and NEITHER
Once you have two probabilities, four rules cover almost every question. Whether you multiply or add depends on whether the events can happen together.
P(A and B) = P(A) × P(B)
Both happen (independent)
For events that do not affect each other, multiply. The chance of two heads in a row is one half times one half.
P(A or B) = P(A) + P(B) − P(A and B)
At least one happens
Add the two, then subtract the overlap so the shared outcomes are not double-counted. If the events are mutually exclusive the overlap is zero, so you just add.
P(not A) = 1 − P(A)
The complement
The chance an event does not happen is one minus the chance it does. The complement is often the easy road to a hard question.
P(A and B) = 0
Mutually exclusive
Some events cannot occur together — a single die cannot land on both 2 and 5. Then the AND probability is zero and OR is a plain sum.
Conditional & joint
When one event depends on another
"What is the probability of A given that B already happened?" is where the simple multiply rule breaks — and where a widget stops helping. Drawing without replacement, screening tests, and any two-stage question all need conditional probability.
P(A|B) = P(A and B) / P(B)
Conditional probability
The chance of A once you know B happened. Divide the joint chance of both by the chance of the condition. This is the rule for "without replacement" draws, where each step shrinks the pool.
two aces without replacement: 4/52 × 3/51
P(A and B) = P(A) × P(B|A)
The general multiplication rule
For dependent events, the joint probability multiplies the first event by the second given the first. When the events are independent P(B|A) = P(B), and this collapses back to the plain product.
1st ace 4/52, 2nd ace 3/51 → 1/221
P(B|A) = P(A|B) × P(B) / P(A)
Bayes' theorem
When you know one direction of a conditional but need the other — a positive test result versus actually having the condition — Bayes rescales it by how common each event is on its own.
flip a known conditional into the one you want
independent: P(A|B) = P(A)
Independence, restated
Events are independent exactly when the condition changes nothing: P(A|B) equals P(A). If knowing B moves the odds of A, they are dependent and the general rules above apply.
a coin flip after a die roll is unchanged
Worked examples
Probability of everyday things
The same favorable-over-total recipe handles coins, dice, cards, and marbles. Switch tabs to see each one computed from scratch.
Flipping heads on a fair coin
Favorable 1 · Total 2
Divide: 1 / 2 = 0.5
As a percent: 50%
P = 1/2 = 0.5 = 50%
Repeated trials
The same event, again and again
"What is the chance of at least one six in three rolls?" is one of the most searched probability questions — and the trick is to flip it around.
P(all) = pⁿ
It happens every time
Multiply the single-trial probability by itself once for each trial. Long strings of success get unlikely fast.
P(none) = (1 − p)ⁿ
It never happens
Raise the failure chance to the power of the number of trials. This is the quantity the next rule subtracts.
P(≥1) = 1 − (1 − p)ⁿ
At least once
The complement of never. Far easier than adding up exactly-one, exactly-two, and so on — and the one people actually want.
Add a 1 inchance row to the table
| Single try | 1 try | 2 tries | 3 tries | 5 tries | 10 tries | 25 tries | 50 tries |
|---|---|---|---|---|---|---|---|
| Coin lands heads | 50% | 75% | 87.5% | 96.9% | 99.9% | 100% | 100% |
| Roll a 6 on a die | 16.7% | 30.6% | 42.1% | 59.8% | 83.8% | 99% | 100% |
| Draw a chosen rank | 7.7% | 14.8% | 21.3% | 33% | 55.1% | 86.5% | 98.2% |
| Roulette single number | 2.7% | 5.3% | 7.9% | 12.8% | 24% | 49.6% | 74.6% |
| 1-in-100 chance | 1% | 2% | 3% | 4.9% | 9.6% | 22.2% | 39.5% |
Each cell is 1 − (1 − p)n — the chance the event happens at least once across that many independent tries.
Dice & cards
Two-dice sums and card draws
The two questions asked most often, answered exactly. Two dice have 36 equally likely ordered outcomes; a shuffled deck has 52 cards. Every probability below is a plain favorable-over-total count.
| Sum | Ways | Probability | Percent |
|---|---|---|---|
| 2 | 1 | 1/36 | 2.78% |
| 3 | 2 | 1/18 | 5.56% |
| 4 | 3 | 1/12 | 8.33% |
| 5 | 4 | 1/9 | 11.11% |
| 6 | 5 | 5/36 | 13.89% |
| 7 | 6 | 1/6 | 16.67% |
| 8 | 5 | 5/36 | 13.89% |
| 9 | 4 | 1/9 | 11.11% |
| 10 | 3 | 1/12 | 8.33% |
| 11 | 2 | 1/18 | 5.56% |
| 12 | 1 | 1/36 | 2.78% |
| Event | Probability | Percent |
|---|---|---|
| A specific card (e.g. ace of spades) | 1/52 = 0.0192 | 1.92% |
| An ace (any suit) | 1/13 = 0.0769 | 7.69% |
| A card of a chosen suit | 1/4 = 0.25 | 25% |
| A face card (J, Q, K) | 3/13 = 0.2308 | 23.08% |
| A red card | 1/2 = 0.5 | 50% |
| A number card 2–10 | 9/13 = 0.6923 | 69.23% |
For two cards in a row, chain the draws: with replacement they stay independent; without replacement the second draw uses the conditional rule above (one fewer card in the deck).
Quick chart
"1 in N" odds as a probability
Convert everyday odds into a decimal, a percentage, and a fraction at a glance. Type any number to drop it into the table.
Look up 1 into add it to the table
| Odds | Decimal | Percent | Fraction |
|---|---|---|---|
| 1 in 2 | 0.5 | 50% | 1/2 |
| 1 in 3 | 0.333333 | 33.333% | 1/3 |
| 1 in 4 | 0.25 | 25% | 1/4 |
| 1 in 5 | 0.2 | 20% | 1/5 |
| 1 in 6 | 0.166667 | 16.667% | 1/6 |
| 1 in 8 | 0.125 | 12.5% | 1/8 |
| 1 in 10 | 0.1 | 10% | 1/10 |
| 1 in 13 | 0.076923 | 7.692% | 1/13 |
| 1 in 20 | 0.05 | 5% | 1/20 |
| 1 in 50 | 0.02 | 2% | 1/50 |
| 1 in 100 | 0.01 | 1% | 1/100 |
| 1 in 1000 | 0.001 | 0.1% | 1/1000 |
| 1 in 1000000 | 0.000001 | 0% | 1/1000000 |
Where it shows up
Why probability matters
The same arithmetic underpins risk, games, and forecasts you meet every day.
genetics & risk
Health and biology
Inheritance ratios, test accuracy, and screening results are all read off the same favorable-over-total ratio.
house edge
Games and betting
Casinos, lotteries, and card games are priced entirely by probability — the gap between true odds and payout is the edge.
P(rain)
Forecasts and quality
Weather forecasts, defect rates, and reliability figures express how often an outcome is expected over many trials.
Worth knowing
Rules and pitfalls to remember
A handful of facts that keep probability questions from going wrong.
Σ P = 1
All outcomes sum to one
The probabilities of every possible outcome must add up to exactly one. If they do not, an outcome is missing or one was counted twice.
complement first
Use the complement
When "at least one" is messy, find the chance of none and subtract from one. It is almost always the shorter path.
× for and · + for or
Multiply vs add
AND multiplies, OR adds. Mixing them up is the single most common mistake — anchor on the word in the question.
with vs without replacement
Does the pool change?
Drawing without replacing changes the total for the next draw, so the events are no longer independent. Recount the pool each step.
rare ≠ impossible
Low is not zero
A one-in-a-million chance still happens given enough trials. Repeated-trial math shows how small risks accumulate.
independent ≠ exclusive
Two different ideas
Mutually exclusive events are highly dependent — one happening rules the other out. Independent events have no effect on each other at all.
FAQ
Common questions
Divide the number of favorable outcomes by the total number of equally likely outcomes. Drawing a heart from a 52-card deck has 13 favorable outcomes out of 52, so P = 13 / 52 = 0.25 = 25%. Every probability lands between 0 (impossible) and 1 (certain).
For a single event, P(event) = favorable outcomes ÷ total outcomes. For two independent events both happening, P(A and B) = P(A) × P(B). For at least one of two events, P(A or B) = P(A) + P(B) − P(A) × P(B). For the same event repeated n times happening at least once, P = 1 − (1 − p)ⁿ.
Independent events do not affect each other — flipping a coin and rolling a die — so both can happen together and you multiply: P(A and B) = P(A) × P(B). Mutually exclusive events cannot both happen at once — rolling a 2 and a 5 on one die — so P(A and B) = 0 and P(A or B) = P(A) + P(B) with no overlap to subtract.
P(A and B) is the chance that both events happen; for independent events it is P(A) × P(B). P(A or B) is the chance that at least one happens; for independent events it is P(A) + P(B) − P(A) × P(B). The subtraction removes the overlap so the shared outcomes are not counted twice.
Find the chance it never happens and subtract from 1. If a single trial has probability p, then over n independent trials the chance of at least one success is 1 − (1 − p)ⁿ. Rolling a 6 at least once in three rolls is 1 − (5/6)³ ≈ 0.4213, or about 42%.
Multiply the decimal probability by 100 and add a percent sign. A probability of 0.25 becomes 0.25 × 100 = 25%. To go the other way, divide the percentage by 100 to get the decimal.
A '1 in N' chance is a probability of 1 ÷ N. A 1-in-4 chance is 1/4 = 0.25 = 25%; a 1-in-100 chance is 0.01 = 1%. The lookup chart on this page converts common odds to decimals, percents, and fractions.
No. A probability is always between 0 and 1 inclusive. A value of 0 means the event cannot happen, 1 means it is certain, and anything in between is its relative chance. A percent therefore runs from 0% to 100%.
Conditional probability is the chance of A once you already know B happened: P(A|B) = P(A and B) ÷ P(B). Drawing two aces from a deck without replacement is P(second ace | first ace) = 3/51, because the first draw changed the pool. It is the tool for 'without replacement' and dependent-event questions.
Two dice have 36 equally likely ordered outcomes. A sum of 7 can happen 6 ways (1-6, 2-5, 3-4, 4-3, 5-2, 6-1), so P(7) = 6/36 = 1/6 ≈ 16.67%. Seven is the most likely total; 2 and 12 are the least, at 1/36 each. The dice table on this page lists every sum.
Use the complement: the chance of no 6 in one roll is 5/6, so in four rolls it is (5/6)⁴ ≈ 0.4823. At least one 6 is 1 − 0.4823 ≈ 0.5177, about 51.8%. The 'at least once' chart on this page does this for any single-trial chance across 1 to 50 tries.
It depends on replacement. With the card put back, the draws are independent: 4/52 × 4/52 = 1/169 ≈ 0.59%. Without replacement they are dependent: 4/52 × 3/51 = 1/221 ≈ 0.45%, because only 3 aces remain in 51 cards.
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Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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