Probability Calculator (Free)

A probability calculator combines the chances of events with a few rules. For a coin landing heads (1/2) and a die showing 6 (1/6), which are independent: P(A and B) = 1/2 × 1/6 = 1/12 ≈ 8.33%, and P(A or B) = 1/2 + 1/6 − 1/12 = 7/12 ≈ 58.33%. A single event is favorable outcomes ÷ total outcomes.

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How to use the probability calculator

1. Two events: choose whether A and B are independent, mutually exclusive or dependent, then enter P(A) and P(B) (and P(A and B) for dependent events). Decimals, fractions and percents all work.
2. Read P(A and B) and P(A or B) at the top, then the full table: exactly one, neither, not A, not B, A but not B, and the conditional probabilities P(A | B) and P(B | A). Tap any row to shade that region of the Venn diagram.
3. At least once: enter the chance on each try and the number of tries to get the chance of at least one success, plus how many tries you need for a 50%, 90% or 99% chance.
4. One event & odds: turn favorable and total outcomes, odds (for or against) or a probability into a fraction, decimal, percent and odds.

The basic probability formula

When every outcome is equally likely, P(event) = number of favorable outcomes ÷ total number of outcomes. A probability is always between 0 (impossible) and 1 (certain), and you can write it as a fraction, a decimal or a percent. Drawing an ace from a 52-card deck: 4 ÷ 52 = 1/13 ≈ 0.0769 ≈ 7.69%. Rolling a sum of 7 with two dice: 6 of the 36 outcomes work, so 6/36 = 1/6.

The complement is the chance the event does not happen: P(not A) = 1 − P(A). The chance of not drawing an ace is 12/13.

Probability rules for two events

Independent, dependent and mutually exclusive events

• Independent: one event does not change the chance of the other, like a coin flip and a die roll. Multiply: P(A and B) = P(A) × P(B).
• Dependent: one event changes the other’s chance, like drawing two cards without putting the first back. The chance of two aces is 4/52 × 3/51 = 1/221 ≈ 0.45%, because after one ace only 3 aces remain among 51 cards.
• Mutually exclusive: the events cannot happen together, like rolling a 1 and rolling a 6 on the same die. Then P(A and B) = 0 and P(A or B) = P(A) + P(B).

Mutually exclusive is not the same as independent. If A and B cannot happen together, learning that A happened tells you B did not, so they are dependent (unless one of them has probability 0).

Conditional probability

P(A | B), read “A given B”, is the chance of A when you already know B happened. You restrict attention to B and ask what share of it is also A: P(A | B) = P(A ∩ B) ÷ P(B). In the Venn diagram this is the overlap as a fraction of circle B.

Example: a card is a face card (12 of 52). What is the chance it is a heart? Three face cards are hearts, so P(heart | face) = (3/52) ÷ (12/52) = 3/12 = 1/4. Here 1/4 equals P(heart), so the two events are independent, which you can confirm with 13/52 × 12/52 = 3/52.

The “at least one” rule

To find the chance that something happens at least once in n independent tries, work with the complement: P(at least once) = 1 − (1 − p)ⁿ. The chance of rolling at least one 6 in 4 rolls is 1 − (5/6)⁴ = 671/1296 ≈ 51.77%. A 1% chance tried 100 times gives 1 − 0.99¹⁰⁰ ≈ 63.40%, not 100%. Multiplying p × n gives the expected number of successes, not a probability. For the chance of exactly k successes, use the binomial distribution.

Odds and probability

Odds compare the two parts of the chances with each other; probability compares one part with the whole. Odds of a to b in favor mean a chances for and b against, so P = a ÷ (a + b). Odds of 3 to 2 in favor give 3/5 = 60%. Odds of 5 to 1 against give 1/6 ≈ 16.67%. Going the other way, a probability of 20% = 1/5 means 1 favorable part and 4 unfavorable parts: odds of 1 to 4 in favor, or 4 to 1 against.

Worked examples

Example 1: a heart or a face card

P(heart) = 13/52 and P(face card) = 12/52, and 3 cards are both. P(heart or face) = 13/52 + 12/52 − 3/52 = 22/52 = 11/26 ≈ 42.31%. Without subtracting the 3 overlapping cards you would count them twice and get 25/52.

Example 2: rain and wind

A forecast gives rain 40%, wind 25% and both 15%. P(rain or wind) = 0.40 + 0.25 − 0.15 = 0.50, so the chance of neither is 50%. P(rain | wind) = 0.15 ÷ 0.25 = 60%. Since 0.40 × 0.25 = 0.10 is not 0.15, the events are dependent: wind makes rain more likely.

Example 3: a famous dice bet

In the 1600s gamblers asked which is more likely: at least one 6 in 4 rolls of one die, or at least one double 6 in 24 rolls of two dice. The first is 1 − (5/6)⁴ ≈ 51.77%. The second is 1 − (35/36)²⁴ ≈ 49.14%. The rule of thumb “24 × 1/36 = 4 × 1/6” makes them look equal, but the at-least-one rule shows they are not.

Common mistakes to avoid

• Adding without subtracting the overlap. P(A or B) = P(A) + P(B) only for mutually exclusive events.
• Multiplying dependent events as if independent. Without replacement, the second chance changes.
• Mixing up P(A | B) and P(B | A). In the card example P(heart | face) = 1/4 but P(face | heart) = 3/13.
• Confusing odds with probability. Odds of 1 to 4 are a probability of 1/5, not 1/4.
• Getting a probability above 1. If your answer is more than 1 or below 0, a rule was used where it does not apply.

Probability practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you calculate probability?

Divide the number of favorable outcomes by the total number of equally likely outcomes. A bag with 3 red and 7 blue marbles gives P(red) = 3/10 = 0.3 = 30%. For two events, use the multiplication rule for "and" and the addition rule for "or".

Is probability calculated in percentage?

It can be. Fractions, decimals and percents are three ways to write the same value: 1/4 = 0.25 = 25%. Textbooks often use fractions or decimals; weather forecasts and polls use percents.

How do you calculate the probability of multiple events?

For independent events that must all happen, multiply their probabilities: three heads in a row is (1/2)³ = 1/8. For "at least one", subtract the chance that none happen from 1. For dependent events, multiply by the conditional probability at each step.

How do you calculate dice probability?

Count the outcomes. One die has 6, two dice have 36. A sum of 7 comes from 6 pairs (1-6, 2-5, 3-4, 4-3, 5-2, 6-1), so its probability is 6/36 = 1/6. Double sixes happen in 1 of 36 rolls.

Is probability exact?

Theoretical probability is exact: a fair coin lands heads with probability 1/2. Experimental probability (results ÷ trials) varies and gets closer to the theoretical value as the number of trials grows, which is the law of large numbers. This calculator keeps fractions exact and shows decimals rounded to 6 places.

Related calculators

• Binomial probability calculator
• Expected value calculator
• Permutations and combinations calculator
• Dice roller
• Coin flip