How do you calculate the probability of A or B?

P(A or B) = P(A) + P(B) - P(A and B). Adding the two probabilities counts the overlap twice, so it is taken away once. With P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2 that is 0.5 + 0.4 - 0.2 = 0.7. For mutually exclusive events the overlap is 0 and the rule becomes a plain sum. The page prints this line with your own numbers in it.

How do you find the probability of A and B?

It depends on how the events relate, which is why the page asks. For independent events P(A and B) = P(A) x P(B), so 0.5 and 0.4 give 0.2. For mutually exclusive events it is 0. From a conditional it is P(A given B) x P(B), or P(B given A) x P(A). If you already know P(A and B), you can type it in directly.

What is the difference between independent and mutually exclusive events?

Independent events do not change each other's chances: P(A given B) equals P(A), and P(A and B) equals P(A) x P(B). Mutually exclusive events cannot happen together: P(A and B) is 0. Two events that each have a probability above 0 cannot be both. When you type in P(A and B) or a conditional, the page checks whether your numbers make A and B independent and says which, with the product it compared against.

How do I calculate conditional probability?

P(A given B) = P(A and B) / P(B): the share of B's probability in which A also happens. With P(A and B) = 0.2 and P(B) = 0.4, P(A given B) is 0.5. It is not defined when P(B) is 0, and the page says undefined rather than dividing by zero. The same entries also give P(B given A), P(A given not B) and P(B given not A).

Why does the calculator say my numbers are impossible?

Two probabilities limit how much the events can overlap. P(A and B) cannot be larger than the smaller of P(A) and P(B), and when P(A) + P(B) is more than 1 the events must overlap by at least P(A) + P(B) - 1. With P(A) = 0.5 and P(B) = 0.4, P(A and B) has to lie between 0 and 0.4, so 0.45 is refused. The message gives the range that would work, and a conditional you type is checked against the same bounds.

What is the probability of something happening at least once?

For independent tries it is 1 minus the chance it never happens: 1 - (1 - p)^n. For p = 0.5 and 2 tries that is 1 - 0.25 = 0.75, and for a 1/6 chance over 3 tries it is 91/216, about 0.421296. The repeat section works this out for A, B, both together or either one, for 1 to 1,000 tries, beside the chance of it happening every time and never.

Can I enter fractions or percentages?

Yes. Each probability can be a decimal such as 0.25, a percentage such as 25% or a fraction such as 1/4, and all three are read as exactly 1/4. Decimals with an exponent, such as 2.5e-7, are read too. Results can be shown as decimals, fractions or percentages. Any value that does not end within twelve decimal places, whether a result or a number inside a line of working, is rounded to six significant figures and marked ≈. The fraction view shows each value exactly, except a fraction longer than 60 characters and a repeat too long for exact fractions, which appear as marked decimals.

Is the Venn diagram drawn to scale?

The two circles have areas in the ratio P(A) to P(B), and the region where they overlap has the area of P(A and B) on the same scale. The box around them stands for everything else and is not to scale, because a circle with the same area as the box would not fit inside it. The diagram downloads as an SVG or a PNG on a white background, with your entries above it and all four region values written underneath.

How do I convert odds to probability?

Choose the odds input format in the converter. Odds in favor of 1:3 mean 1 favorable outcome among 4, so p = 1/(1+3) = 25%. Odds against of 3:1 describe the same chance. Decimal odds of 4, fractional odds of 3/1 and American odds of +300 also imply 25%. These conversions use exact fractions and show the equivalent formats.

What is implied probability?

It is the probability represented by stated odds. For decimal odds D, p = 1/D: 2.5 implies 40%. For fractional odds a/b, p = b/(a+b). Positive American odds +A imply 100/(A+100); negative odds -A imply A/(A+100), so -150 implies 60%. A quoted price may include a margin; the page does not remove it or estimate the true chance of a real outcome.

Can I turn a probability into odds?

Yes. Choose Probability and enter a decimal, percent or fraction. At 25%, odds in favor are 1:3, odds against are 3:1, decimal odds are 4 and fractional odds are 3/1. At 0%, decimal and fractional odds have no finite value. American odds have no finite standard value at 0% or 100%; the page labels those endpoints instead of dividing by zero.

Can I use this on a phone or in a browser?

The same probability and odds controls work in the page on a phone, tablet or desktop browser. Type a decimal point, fraction slash or ratio colon as needed. There is no download or account step; copying or downloading a diagram depends on the permissions available in your browser.

Does this predict medical, financial or sports outcomes?

No. The calculator only derives consequences of probabilities or odds you enter. It has no patient data, market model, team ratings or weather model, so it cannot estimate pregnancy, cancer, default, stock-option or match-outcome probabilities.

How do I calculate the probability of multiple events?

Use Multiple independent events and enter each probability separately, then confirm independence. The probability of all events is the product of their probabilities; at least one is 1 minus the product of their complements. For 0.5, 0.4 and 0.2, all three is 0.04, none is 0.24 and at least one is 0.76. Dependent events need additional joint information and cannot be inferred from this list.

How do combinations give a probability?

Use Combinations and probability to count unordered selections with C(n, k), then enter the number of favorable combinations and confirm that all combinations are equally likely. There are C(5, 2) = 10 pairs from five items; if three pairs are favorable, the chance is 3/10 = 30%. This counts selections without replacement and without order. It does not enumerate which pairs meet your event.

Event Probability Calculator

Enter two probabilities and choose how the events relate to get unions, intersections and conditionals with formulas and a Venn diagram. You can also convert stated odds, combine a list of independent event probabilities, or use n choose k to find a probability from equally likely combinations.

eventsA and B

P(A)

P(B)

Type each probability as a decimal (0.25), a percentage (25%) or a fraction (1/4).

How A and B relate

Type the probability that A and B both happen.

P(A and B)

Independent tries

Event repeated

Each try repeats the whole experiment, independently of the others, with the same probabilities.

Feasible P(A and B) for these entries

from max(0, 0.5 + 0.4 - 1) = 0 to min(0.5, 0.4) = 0.4

Venn diagram

Circles A and B overlap. A only 0.3, A and B 0.2, B only 0.2, outside both 0.3, total 1. The circles and their overlap are drawn to scale with each other; the box around them stands for everything else and is not to scale.

resultsfrom your entries, ≈ marks rounding

P(A and B): 0.2, entered
P(A or B): 0.5 + 0.4 - 0.2 = 0.7
P(not A): 1 - 0.5 = 0.5
P(not B): 1 - 0.4 = 0.6
P(A given B): 0.2 / 0.4 = 0.5
P(B given A): 0.2 / 0.5 = 0.4
A only: 0.5 - 0.2 = 0.3
B only: 0.4 - 0.2 = 0.2
Neither: 1 - 0.7 = 0.3
Exactly one: 0.7 - 0.2 = 0.5
P(A given not B): 0.3 / 0.6 = 0.5
P(B given not A): 0.2 / 0.5 = 0.4

A at least once in 2 tries: 1 - (1 - 0.5)^2 = 1 - 0.25 = 0.75
A in all 2 tries: (0.5)^2 = 0.25
A in none of the 2 tries: (1 - 0.5)^2 = 0.25

With these entries A and B are independent: P(A and B) = 0.2 equals P(A) x P(B) = 0.5 x 0.4 = 0.2.

Binomial, normal and Poisson questions have their own calculators: binomial, normal and Poisson.

Show as

P(A and B)

0.2

P(A or B)

0.7

P(A given B)

0.5

Neither

0.3

P(A) 0.5, P(B) 0.4independent, selected
P(A and B) = 0.5 x 0.4 = 0.2; P(A or B) = 0.5 + 0.4 - 0.2 = 0.7
P(A and B) entered as 0.45P(A) 0.5, P(B) 0.4
rejected: P(A and B) = 0.45 cannot be larger than P(B) = 0.4, because A and B together cannot happen more often than B alone. For these entries P(A and B) runs from 0 to 0.4.
At least one success in 2 triesindependent, p = 0.5 each
1 - (1 - 0.5)^2 = 1 - 0.25 = 0.75

Odds and implied probability

Convert stated odds or a probability. Implied probability is arithmetic on the quoted price; it predicts no outcome and does not remove a margin.

Input format

Probability

Enter 0.25, 25% or 1/4.

Probability: 1/4 = 25%
Odds in favor: 1:3
Odds against: 3:1
Decimal odds: 4
Fractional odds: 3/1
American odds: +300

Multiple independent events

Event probabilities

Enter 2 to 100 probabilities as decimals, percentages or fractions, separated by commas, semicolons or new lines. These can be different events; they need not have the same probability.

These events are independent

Confirm that these events are independent to combine their probabilities. Marginal probabilities alone cannot determine dependent events.

Combinations and probability

Count unordered selections with n choose k, then enter how many combinations meet your event. Probability is favorable combinations divided by all combinations, when every combination is equally likely. This does not count repeated selections or ordered arrangements.

Available items n

Items selected k

Favorable combinations

Each combination is equally likely

Confirm that each combination is equally likely before calculating the probability.

Accuracy. Every figure is exact fraction arithmetic on your entries, and a result that needs rounding for display is shown to six significant figures and marked ≈; a repeat count too long for exact fractions is worked in floating point and marked ≈ the same way. The page describes your assumptions and predicts nothing about real events: A and B are treated as independent only when you select it, and entries that cannot all hold together are refused with the range that would work. Binomial, normal and Poisson questions belong to the distribution calculators linked under the results.

Common questions

How do you calculate the probability of A or B?: P(A or B) = P(A) + P(B) - P(A and B). Adding the two probabilities counts the overlap twice, so it is taken away once. With P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2 that is 0.5 + 0.4 - 0.2 = 0.7. For mutually exclusive events the overlap is 0 and the rule becomes a plain sum. The page prints this line with your own numbers in it.
How do you find the probability of A and B?: It depends on how the events relate, which is why the page asks. For independent events P(A and B) = P(A) x P(B), so 0.5 and 0.4 give 0.2. For mutually exclusive events it is 0. From a conditional it is P(A given B) x P(B), or P(B given A) x P(A). If you already know P(A and B), you can type it in directly.
What is the difference between independent and mutually exclusive events?: Independent events do not change each other's chances: P(A given B) equals P(A), and P(A and B) equals P(A) x P(B). Mutually exclusive events cannot happen together: P(A and B) is 0. Two events that each have a probability above 0 cannot be both. When you type in P(A and B) or a conditional, the page checks whether your numbers make A and B independent and says which, with the product it compared against.
How do I calculate conditional probability?: P(A given B) = P(A and B) / P(B): the share of B's probability in which A also happens. With P(A and B) = 0.2 and P(B) = 0.4, P(A given B) is 0.5. It is not defined when P(B) is 0, and the page says undefined rather than dividing by zero. The same entries also give P(B given A), P(A given not B) and P(B given not A).
Why does the calculator say my numbers are impossible?: Two probabilities limit how much the events can overlap. P(A and B) cannot be larger than the smaller of P(A) and P(B), and when P(A) + P(B) is more than 1 the events must overlap by at least P(A) + P(B) - 1. With P(A) = 0.5 and P(B) = 0.4, P(A and B) has to lie between 0 and 0.4, so 0.45 is refused. The message gives the range that would work, and a conditional you type is checked against the same bounds.
What is the probability of something happening at least once?: For independent tries it is 1 minus the chance it never happens: 1 - (1 - p)^n. For p = 0.5 and 2 tries that is 1 - 0.25 = 0.75, and for a 1/6 chance over 3 tries it is 91/216, about 0.421296. The repeat section works this out for A, B, both together or either one, for 1 to 1,000 tries, beside the chance of it happening every time and never.
Can I enter fractions or percentages?: Yes. Each probability can be a decimal such as 0.25, a percentage such as 25% or a fraction such as 1/4, and all three are read as exactly 1/4. Decimals with an exponent, such as 2.5e-7, are read too. Results can be shown as decimals, fractions or percentages. Any value that does not end within twelve decimal places, whether a result or a number inside a line of working, is rounded to six significant figures and marked ≈. The fraction view shows each value exactly, except a fraction longer than 60 characters and a repeat too long for exact fractions, which appear as marked decimals.
Is the Venn diagram drawn to scale?: The two circles have areas in the ratio P(A) to P(B), and the region where they overlap has the area of P(A and B) on the same scale. The box around them stands for everything else and is not to scale, because a circle with the same area as the box would not fit inside it. The diagram downloads as an SVG or a PNG on a white background, with your entries above it and all four region values written underneath.
How do I convert odds to probability?: Choose the odds input format in the converter. Odds in favor of 1:3 mean 1 favorable outcome among 4, so p = 1/(1+3) = 25%. Odds against of 3:1 describe the same chance. Decimal odds of 4, fractional odds of 3/1 and American odds of +300 also imply 25%. These conversions use exact fractions and show the equivalent formats.
What is implied probability?: It is the probability represented by stated odds. For decimal odds D, p = 1/D: 2.5 implies 40%. For fractional odds a/b, p = b/(a+b). Positive American odds +A imply 100/(A+100); negative odds -A imply A/(A+100), so -150 implies 60%. A quoted price may include a margin; the page does not remove it or estimate the true chance of a real outcome.
Can I turn a probability into odds?: Yes. Choose Probability and enter a decimal, percent or fraction. At 25%, odds in favor are 1:3, odds against are 3:1, decimal odds are 4 and fractional odds are 3/1. At 0%, decimal and fractional odds have no finite value. American odds have no finite standard value at 0% or 100%; the page labels those endpoints instead of dividing by zero.
Can I use this on a phone or in a browser?: The same probability and odds controls work in the page on a phone, tablet or desktop browser. Type a decimal point, fraction slash or ratio colon as needed. There is no download or account step; copying or downloading a diagram depends on the permissions available in your browser.
Does this predict medical, financial or sports outcomes?: No. The calculator only derives consequences of probabilities or odds you enter. It has no patient data, market model, team ratings or weather model, so it cannot estimate pregnancy, cancer, default, stock-option or match-outcome probabilities.
How do I calculate the probability of multiple events?: Use Multiple independent events and enter each probability separately, then confirm independence. The probability of all events is the product of their probabilities; at least one is 1 minus the product of their complements. For 0.5, 0.4 and 0.2, all three is 0.04, none is 0.24 and at least one is 0.76. Dependent events need additional joint information and cannot be inferred from this list.
How do combinations give a probability?: Use Combinations and probability to count unordered selections with C(n, k), then enter the number of favorable combinations and confirm that all combinations are equally likely. There are C(5, 2) = 10 pairs from five items; if three pairs are favorable, the chance is 3/10 = 30%. This counts selections without replacement and without order. It does not enumerate which pairs meet your event.

Every figure is exact fraction arithmetic on your entries, and a result that needs rounding for display is shown to six significant figures and marked ≈; a repeat count too long for exact fractions is worked in floating point and marked ≈ the same way. The page describes your assumptions and predicts nothing about real events: A and B are treated as independent only when you select it, and entries that cannot all hold together are refused with the range that would work. Binomial, normal and Poisson questions belong to the distribution calculators linked under the results.