Probability Calculator
Free probability calculator: find the chance of an event as a fraction, decimal, or percent, with rules for independent, dependent, and combined events.
There was an error with your calculation.
| Result | ||
|---|---|---|
| Probability of A NOT occuring: P(A') | 0.5 | |
| Probability of B NOT occuring: P(B') | 0.6 | |
| Probability of A and B both occuring: P(A∩B) | 0.2 | |
| Probability that A or B or both occur: P(A∪B) | 0.7 | |
| Probability that A or B occurs but NOT both: P(AΔB) | 0.5 | |
| Probability of neither A nor B occuring: P((A∪B)') | 0.3 | |
| Probability of A occuring but NOT B: | 0.3 | |
| Probability of B occuring but NOT A: | 0.2 | |
Probability
Probability of A: P(A) = 0.5
Probability of B: P(B) = 0.4
Probability of A NOT occuring: P(A') = 1 - P(A) = 0.5
Probability of B NOT occuring: P(B') = 1 - P(B) = 0.6
Probability of A and B both occuring: P(A∩B) = P(A) × P(B) = 0.2
Probability that A or B or both occur: P(A∪B) = P(A) + P(B) - P(A∩B) = 0.7
Probability that A or B occurs but NOT both: P(AΔB) = P(A) + P(B) - 2P(A∩B) = 0.5
Probability of neither A nor B occuring: P((A∪B)') = 1 - P(A∪B) = 0.3
Probability of A occuring but NOT B: P(A) × (1 - P(B)) = 0.3
Probability of B occuring but NOT A: (1 - P(A)) × P(B) = 0.2
Probability
Probability of A occuring 5 time(s) = 0.65 = 0.07776
Probability of A NOT occuring = (1-0.6)5 = 0.01024
Probability of A occuring = 1-(1-0.6)5 = 0.98976
Probability of B occuring 3 time(s) = 0.33 = 0.027
Probability of B NOT occuring = (1-0.3)3 = 0.343
Probability of B occuring = 1-(1-0.3)3 = 0.657
Probability of A occuring 5 time(s) and B occuring 3 time(s) = 0.65 × 0.33 = 0.00209952
Probability of neither A nor B occuring = (1-0.6)5 × (1-0.3)3 = 0.00351232
Probability of both A and B occuring = (1-(1-0.6)5) × (1-(1-0.3)3) = 0.65027232
Probability of A occuring 5 times but not B = 0.65 × (1-0.3)3 = 0.02667168
Probability of B occuring 3 times but not A = (1-0.6)5 × 0.33 = 2.7648e-4
Probability of A occuring but not B = (1-(1-0.6)5) × (1-0.3)3 = 0.33948768
Probability of B occuring but not A = (1-0.6)5 × (1-(1-0.3)3) = 0.00672768
Probability
The probability between -1 and 1 is 0.68268
The probability outside of -1 and 1 is 0.31732
The probability of -1 or less (≤-1) is 0.15866
The probability of 1 or more (≥1) is 0.15866
| CONFIDENCE INTERVALS TABLE | ||
|---|---|---|
| CONFIDENCE | RANGE | N |
| 0.6828 | -1.00000 – 1.00000 | 1 |
| 0.8 | -1.28155 – 1.28155 | 1.281551565545 |
| 0.9 | -1.64485 – 1.64485 | 1.644853626951 |
| 0.95 | -1.95996 – 1.95996 | 1.959963984540 |
| 0.98 | -2.32635 – 2.32635 | 2.326347874041 |
| 0.99 | -2.57583 – 2.57583 | 2.575829303549 |
| 0.995 | -2.80703 – 2.80703 | 2.807033768344 |
| 0.998 | -3.09023 – 3.09023 | 3.090232306168 |
| 0.999 | -3.29053 – 3.29053 | 3.290526731492 |
| 0.9999 | -3.89059 – 3.89059 | 3.890591886413 |
| 0.99999 | -4.41717 – 4.41717 | 4.417173413469 |
Probability calculator at a glance#
A probability calculator works out how likely an event is. Probability is the number of favorable outcomes divided by the total number of possible outcomes, so P = favorable / total. The result runs from 0 (impossible) to 1 (certain), and you can read it as a fraction, a decimal, or a percentage.
For example, rolling a 4 on a six-sided die has 1 favorable outcome out of 6, so P = 1/6 which is about 0.167, or 16.7%.
| Outcome On One die | Fraction | Decimal | Percent |
|---|---|---|---|
| Rolling a 4 | 1/6 | 0.167 | 16.7% |
| Rolling an even number | 3/6 | 0.5 | 50% |
| Rolling 4 or higher | 3/6 | 0.5 | 50% |
| Rolling a number below 5 | 4/6 | 0.667 | 66.7% |
For two independent events, where one does not affect the other, multiply their probabilities. Rolling a 4 twice in a row is 1/6 times 1/6 = 1/36, which is about 0.028, or 2.8%. For dependent events the second probability changes once the first has happened.
Enter your favorable and total outcomes in the calculator above for the exact probability. Results are rounded for display, so a repeating decimal like 0.1666... shows as 0.167.
Theoretical vs empirical probability#
Theoretical probability comes from the structure of the problem when every outcome is equally likely: it is favorable outcomes divided by total outcomes. Rolling a 4 on a fair die is 1/6. Empirical probability comes from observed data: the number of times an event happened divided by the number of trials. If rain fell on 90 of the last 180 days, the empirical probability of rain is 90/180 = 0.5. The two converge as the number of trials grows.
Independent and dependent events#
Two events are independent when one does not change the other; for these you multiply. Rolling two 4s in a row is 1/6 times 1/6 = 1/36. Events are dependent when the first changes the second, which happens when you sample without replacement. Drawing two aces from a 52-card deck is 4/52 times 3/51 = 12/2652 = 1/221, because after the first ace there are 3 aces left in 51 cards.
Mutually exclusive events and the addition rule#
Two events are mutually exclusive when they cannot both happen at once, like a single die showing both even and odd. For mutually exclusive events you add: the probability of rolling a 1 or a 2 is 1/6 + 1/6 = 2/6 = 1/3. When events can overlap, subtract the shared part to avoid counting it twice: P(A or B) = P(A) + P(B) minus P(A and B).
The complement: probability of "not"#
The probability that an event does not happen is 1 minus the probability that it does. If rolling a 4 is 1/6, then not rolling a 4 is 1 minus 1/6 = 5/6. The complement is often the quickest route to "at least one" problems: the chance of at least one 6 in two rolls is 1 minus the chance of no 6, which is 1 minus (5/6 times 5/6) = 1 minus 25/36 = 11/36.
Frequently asked questions#
How do you calculate probability?#
Divide the number of favorable outcomes by the total number of possible outcomes. Rolling a 4 on a six-sided die is 1 favorable out of 6 total, so 1/6, about 0.167 or 16.7%.
What is the difference between independent and dependent events?#
Independent events do not affect each other (two separate die rolls), and you multiply their probabilities. Dependent events do (drawing cards without replacement), and the second probability changes after the first event.
How does the multiplication rule work?#
For two independent events, the probability that both happen is the product of their separate probabilities. If P(A) = 0.5 and P(B) = 0.3, then P(A and B) = 0.5 times 0.3 = 0.15.
What are mutually exclusive events?#
Events that cannot occur at the same time, such as a single die landing on both an even and an odd number. Their probabilities add: P(A or B) = P(A) + P(B).
What is the difference between theoretical and experimental probability?#
Theoretical probability is calculated from equally likely outcomes without running a trial. Experimental (empirical) probability is measured from real trials: favorable results divided by the number of trials.
What range can a probability take?#
From 0 to 1. A probability of 0 means impossible, 1 means certain, and 0.5 means an even chance. You can express the same value as a fraction, a decimal, or a percentage.
Related calculators
View all calculators
Rounding CalulatorFree rounding calculator. Round any number to the nearest 10, 100, whole number, decimal place, or cent using the round-half-up rule, with examples.
Scientific Notation CalculatorFree scientific notation calculator: convert numbers to and from a x 10^n, plus e-notation and engineering notation, with worked steps.
Sig Fig CalculatorFree sig fig calculator: count the significant figures in any number and round to a chosen count, with the four sig fig rules and examples.
Significant Figures CalculatorFree significant figures calculator: round to a set number of sig figs and apply the add, subtract, multiply and divide precision rules.
Simplifying Fractions CalculatorFree simplifying fractions calculator that reduces any fraction to lowest terms, shows the GCF, the steps, and the mixed-number form.
Slope CalculatorFree slope calculator: find the slope of a line from two points with m = (y2-y1)/(x2-x1), plus the line equation, grade percentage and angle.