Probability Calculator
Work out the chance of one event from favorable and total outcomes, combine two independent events with AND or OR, or find the probability that an event does not happen. Enter probabilities as decimals, percentages or fractions, and see every formula worked through with your numbers. Free, no sign-up.
Single event
P = 0.1667 (16.6667%)
P(event) = favorable ÷ total = 1 ÷ 6 = 0.1667
As a percentage: 0.1667 × 100 = 16.6667%
Odds in favor: 1 favorable to 5 unfavorable = 1 : 5
Two independent events — AND / OR
P(A and B) = 0.0833 (8.3333%)
P(A and B) = P(A) × P(B)
= 0.5 × 0.1667 = 0.0833
P(A or B) = 0.5833 (58.3333%)
P(A or B) = P(A) + P(B) − P(A) × P(B)
= 0.5 + 0.1667 − 0.0833 = 0.5833
Honest note: these formulas assume A and B are independent — one happening does not change the other's probability. For dependent events (like drawing cards without replacement) they do not apply.
Complement — P(not A)
P(not A) = 0.7 (70%)
P(not A) = 1 − P(A)
= 1 − 0.3 = 0.7
How to Use the Probability Calculator
- 1. Single event. Enter the number of favorable outcomes and the total number of equally likely outcomes (whole numbers, favorable no larger than total). You get the probability as a decimal and a percentage, plus the odds in favor.
- 2. Two events. Enter P(A) and P(B) as a decimal (0.5), a percentage (50%) or a fraction (1/2). You get P(A and B) and P(A or B) for independent events. A bare number above 1 is read as a percentage, so 2 means 2%, not 200%.
- 3. Complement. Enter P(A) in the same formats and get P(not A) = 1 − P(A).
- 4. Read the working. Each result shows its formula and the arithmetic, rounded to four decimals.
The Formulas
- Single event: P = favorable ÷ total. Rolling a 6 on a fair die is 1 ÷ 6 = 0.1667, or 16.67%, with odds of 1 : 5 in favor.
- Independent AND: P(A and B) = P(A) × P(B). Both events must happen, so the chances multiply.
- Independent OR: P(A or B) = P(A) + P(B) − P(A) × P(B). Adding counts the overlap twice, so one copy is subtracted.
- Complement: P(not A) = 1 − P(A).
A coin comes up heads (50%) and a die shows a 6 (1/6)
P(A and B) = 0.5 × 0.1667 = 0.0833, about 8.33%
P(A or B) = 0.5 + 0.1667 − 0.0833 = 0.5833, about 58.33%
P(not B) = 1 − 0.1667 = 0.8333, about 83.33%
"At Least One" Questions
The complement is the fastest route to "at least one": P(at least one) = 1 − P(none). For two rolls of a die and at least one 6, the chance of no six on either roll is 5/6 × 5/6 = 0.6944, so the answer is 1 − 0.6944 = 0.3056, about 30.56%. Entering 1/6 and 1/6 into the two-event section and reading P(A or B) gives the same 0.3056. For three or more rolls, multiply the "no six" chance repeatedly and take the complement.
When the Formulas Do Not Apply
- Dependent events. Drawing two cards without putting the first back changes the second probability, so multiplying the original chances is wrong. The AND and OR lines here assume independence; use conditional probability or the Hypergeometric Distribution Calculator instead.
- Mutually exclusive events. If A and B cannot both happen, such as rolling a 2 or a 5 on one die, P(A or B) is simply P(A) + P(B). The overlap term is zero, and this calculator's OR line would understate the answer if you entered the two chances as if they were independent.
- Unequal outcomes. The single-event formula assumes every outcome is equally likely. A loaded die or an uneven spinner needs the individual probabilities.
- Probability versus odds. A 1-in-6 chance is a probability of 1/6 and odds of 1 : 5 in favor; they are not the same number.
Which Tool Do You Need?
| You want | Use |
|---|---|
| Basic event, AND, OR and not probabilities | This Probability Calculator |
| How many ways to arrange or choose items | Permutation and Combination Calculator |
| Chances of k successes in n trials | Binomial Distribution Calculator |
| Updating a probability when you see evidence | Bayes' Theorem Calculator |
| The average result over many trials | Expected Value Calculator |
Frequently Asked Questions
- How do I calculate the probability of a single event?
- Divide the number of favorable outcomes by the total number of equally likely outcomes. A fair die landing on 6 is 1 ÷ 6 ≈ 0.1667, or about 16.67%.
- How do I calculate the probability of two events both happening?
- For independent events, multiply: P(A and B) = P(A) × P(B). Two fair coin flips both landing heads is 0.5 × 0.5 = 0.25.
- How do I calculate the probability of A or B?
- For independent events, P(A or B) = P(A) + P(B) − P(A) × P(B). The subtraction removes the overlap where both happen so it is not counted twice. For mutually exclusive events there is no overlap, and you just add.
- What is a complement in probability?
- The complement of an event is its not happening: P(not A) = 1 − P(A). If rain has a 30% chance, no rain has 70%. It is the shortcut for "at least one" problems.
- What does "independent events" mean?
- Two events are independent when one happening does not change the probability of the other, like two separate coin flips. The AND and OR formulas here give wrong answers for dependent events, such as drawing cards without replacement.
- What is the difference between probability and odds?
- Probability compares favorable outcomes with all outcomes; odds compare favorable with unfavorable. A 1-in-6 chance is a probability of 1/6 and odds of 1 : 5 in favor.