Every two-event result
Get A and B, A or B, exactly one, neither, both complements and both conditional probabilities at once.
Events · Binomial · Odds
Probability Calculator
This free probability calculator works out the chance of two events together or apart, the odds of k successes in repeated trials, and conversions between odds and probability. Enter numbers as decimals, percentages or fractions and see every answer instantly.
- P(A and B), P(A or B)
- Live Venn diagram
- Binomial with chart
- Odds converter
- Accepts 1/6, 25%, 0.3
Calculate probability
= 50%
= 16.67%
Try an example
Enter probabilities as decimals (0.25), percentages (25%) or fractions (1/4).
Probability of A and B
8.33%
About 1 in 12 · 0.0833
All results
P(A∩B)A and B both happen8.33%1 in 12P(A∪B)A or B, or both58.33%1 in 1.71P(A⊕B)Exactly one happens50%1 in 2P(neither)Neither happens41.67%1 in 2.4P(A′)A does not happen50%1 in 2P(B′)B does not happen83.33%1 in 1.2P(A|B)A, given B happened50%1 in 2P(B|A)B, given A happened16.67%1 in 6
= 50%
Try an example
Use this for any yes-or-no event repeated a fixed number of times, where each trial has the same chance and does not affect the others.
Probability of exactly 5 successes
24.61%
About 1 in 4.06 · 0.2461
Distribution of successes
Cumulative results
P(X<k)Fewer than k37.7%P(X≤k)At most k62.3%P(X>k)More than k37.7%P(X≥k)At least k62.3%P(X≥1)At least one success99.9%
- Expected value
- 5
- Standard deviation
- 1.581
- Variance
- 2.5
- Most likely
- 5
= 25%
Odds in favor (for : against)
Try an example
Type in either field. Odds of 1 : 3 mean one way to win for every three ways to lose.
Probability
25%
About 1 in 4 · 0.25
- Odds in favor
- 1 : 3
- Odds against
- 3 : 1
- Decimal odds
- 4.00
- Fractional odds
- 3/1
- American odds
- +300
- Chance it fails
- 75%
Features
Why use this probability calculator?
Most probability tools solve one narrow problem and show a bare number. This one covers the three most common questions and shows the answer in the form you need.
Any number format
Type 0.25, 25% or 1/4. The calculator reads all three and shows how it understood your input.
Binomial with a chart
See the full distribution of successes, with your k highlighted, plus the mean and standard deviation.
Live Venn diagram
Watch the diagram split into only A, both, only B and neither as you change the numbers.
Odds made simple
Convert between probability, odds for and against, and decimal, fractional and American odds.
“1 in N” for every answer
Each result also shows a plain-language “1 in N” chance, which is easier to picture than a small decimal.
How it works
How to use the probability calculator
Pick the tab that matches your question, then follow these steps.
Choose a mode
Use Two events for A and B, Binomial for repeated trials, or Odds for conversions.
Enter the chances
Type each probability as a decimal, a percentage or a fraction such as 1/6.
Set the relationship
Say whether A and B are independent, mutually exclusive or overlap by a known amount.
Read and share
Results update as you type. Copy them or share a link that keeps your inputs.
The formulas
Probability formulas explained
Probability is a number from 0 to 1 that measures how likely an event is. These six rules cover most everyday problems, and the calculator uses each of them.
A and B
P(A and B) = P(A) × P(B)This works when the events are independent. For example, two sixes on two dice is 1/6 × 1/6 = 1/36, or 2.78%.
A or B
P(A or B) = P(A) + P(B) − P(A and B)You subtract the overlap so it is not counted twice. If the events cannot happen together, the overlap is zero.
Not A
P(not A) = 1 − P(A)The chance something does not happen is one minus the chance it does. A 30% chance of rain means a 70% chance of none.
A given B
P(A|B) = P(A and B) ÷ P(B)This is the chance of A once you already know B has happened. For independent events, it simply equals P(A).
The “at least once” trick
P(at least 1) = 1 − (1 − p)nThe chance of at least one 6 in four rolls is 1 − (5/6)4, which is about 51.77%.
Exactly k successes
P(X = k) = C(n, k) × pk × (1 − p)n−kC(n, k) counts the ways to choose k of n trials. Exactly 5 heads in 10 flips is 252/1024, or 24.61%.
Reference
Common probabilities at a glance
These everyday examples help you get a feel for how likely different events really are.
| Event | Probability | 1 in N |
|---|---|---|
| Coin lands heads | 50% | 1 in 2 |
| Rolling a 6 on one die | 16.67% | 1 in 6 |
| Rolling two 6s | 2.78% | 1 in 36 |
| Drawing an ace | 7.69% | 1 in 13 |
| Drawing a heart | 25% | 1 in 4 |
| Shared birthday in a group of 23 | 50.73% | About 1 in 2 |
| Royal flush in 5 cards | 0.000154% | 1 in 649,740 |
- Independent events do not affect each other, like two separate coin flips. Multiply to get both.
- Mutually exclusive events cannot happen together, like rolling a 2 and a 5 on one die. Add to get either.
- Probability vs odds. A 25% probability means odds of 1 : 3, because there is one success for every three failures.
- The gambler’s fallacy. Past results do not change independent odds. After five heads, the next flip is still 50%.
Benefits
Where probability helps in real life
Probability is the language of risk. Knowing the numbers leads to calmer, better decisions.
Homework and exams
Check answers for statistics, maths and science classes, and see which rule each question needs.
Games and puzzles
Work out dice, card and board game chances before you make your next move.
Business decisions
Estimate conversion rates, defect rates and the chance a campaign hits its target.
Everyday risk
Put weather forecasts, test results and safety statistics into perspective with “1 in N” figures.
FAQ
Probability calculator FAQs
Clear answers to the questions people ask most about probability.
How do you calculate probability?
Divide the number of ways an event can happen by the total number of equally likely outcomes. For example, the chance of rolling a 6 is 1 favourable outcome out of 6, so 1/6, or about 16.67%.
What is the difference between P(A and B) and P(A or B)?
P(A and B) is the chance both events happen, so it is always the smaller number. P(A or B) is the chance at least one of them happens, so it is always the larger number.
What does independent mean in probability?
Two events are independent when one does not change the chance of the other. Coin flips and dice rolls are independent. Drawing two cards without putting the first one back is not.
What is binomial probability?
It is the chance of getting exactly k successes in n independent trials, where each trial has the same success chance p. Flipping a coin 10 times and counting heads is a classic example.
How do I find the probability of at least one success?
Work out the chance of zero successes, then subtract it from 1. With n tries at chance p, the answer is 1 − (1 − p)n. The Binomial tab shows it automatically.
How do I convert odds to probability?
For odds of a : b in favour, the probability is a ÷ (a + b). So odds of 1 : 3 give 1 ÷ 4, which is 25%. The Odds tab converts in both directions.
Can a probability be more than 1 or 100%?
No. Every probability sits between 0, which means impossible, and 1, which means certain. If a result goes above 1, a step has been added twice or the events were not independent.
What is conditional probability?
It is the chance of one event given that another has already happened, written P(A|B). You find it by dividing P(A and B) by P(B).
Turn uncertainty into clear numbers
From dice games to data analysis, this probability calculator gives you exact answers in seconds. Try your own numbers, then explore our other maths and statistics tools below.
Open the calculatorResults assume the inputs you enter are accurate. Real-world events are not always independent, so choose the relationship that fits your situation.