Probability Calculator

Calculate basic, conditional, and Bayesian probability with odds, an independence check, and a Bayes breakdown.

Probability Calculator tool

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Example

Input: Rolling a 6 on a fair die: favorable outcomes 1, total outcomes 6

Result: P(A) = 0.166667 (16.67%), P(not A) = 0.833333 (83.33%), odds for 1 : 5

P(A) = 1 / 6 = 0.166667. The complement is 1 - 1/6 = 5/6, so the odds are 1 success to 5 failures.

Everyday Probabilities and Odds

EventFavorable of totalP(A)Odds for
Heads on a coin flip1 of 20.5 (50.00%)1 : 1
A 6 on one die1 of 60.166667 (16.67%)1 : 5
An even number on one die3 of 60.5 (50.00%)3 : 3
Drawing an ace from 52 cards4 of 520.076923 (7.69%)4 : 48
Drawing a heart from 52 cards13 of 520.25 (25.00%)13 : 39
A given number, 1 to 101 of 100.1 (10.00%)1 : 9
Two heads from two flips (1 of 4 outcomes)1 of 40.25 (25.00%)1 : 3

Results from the Basic Probability tab. Odds are shown unreduced, as the tool shows them (4 : 48 is the same as 1 : 12).

  1. Select the probability tab you want to use - Basic Probability, Conditional, or Bayesian.
  2. For basic probability, enter the number of favorable outcomes and the total number of outcomes to get P(A), P(not A), and the odds.
  3. For conditional probability, provide P(A), P(B), and P(A and B) to compute P(A given B), P(B given A), and an independence check.
  4. For Bayesian probability, enter the prior P(A), the likelihood P(B|A), and P(B|not A) to compute the posterior.
  5. Review the results as decimals and percentages, and on the Bayesian tab follow the formula breakdown with your numbers.

Basic, Conditional and Bayesian Probability

Probability is a number from 0 to 1 that says how likely something is. A probability of 0 means it cannot happen, and 1 means it is certain. This calculator has three tabs. Basic finds the chance of one event from favorable and total outcomes. Conditional finds the chance of one event given that another has happened. Bayesian updates a probability after new evidence, using Bayes' theorem.

On the Basic tab, the chance of rolling a 6 is 1 favorable outcome out of 6, which is 0.1667, or 16.67%. The tab also shows the chance it does not happen and the odds. Odds compare successes with failures, so rolling a 6 has odds of 1 : 5 for and 5 : 1 against. The odds are not reduced, so drawing an ace from a deck shows 4 : 48.

The Conditional tab uses P(A|B) = P(A and B) / P(B), where P(A|B) means the chance of A given B. With the starting values of 0.4, 0.5 and 0.2, P(A|B) is 0.4, the same as P(A). Knowing B tells you nothing new about A, so the tool reports the events as independent. It also checks that your numbers can coexist. The joint probability cannot be larger than either event, and the chance of A or B cannot pass 1.

The Bayesian tab shows why base rates matter. Its default numbers describe a test for a condition that 1% of people have. The test catches 90% of true cases but also flags 5% of people without it. After a positive result, the chance of having the condition is only 15.38%, because false alarms from the large healthy group outnumber the true cases. A step-by-step breakdown shows the formula with your numbers filled in. For real test results, ask a doctor what they mean for you.

To count outcomes before you divide, use the Combination and Permutation Calculator.

Formula / How It Works

P(A) = favorable / total | P(A|B) = P(A and B) / P(B) | Bayes: P(H|E) = P(E|H) x P(H) / P(E)

Frequently Asked Questions

Basic probability is the chance of an event with no extra information: favorable outcomes divided by total outcomes. Conditional probability is the chance of an event once you know another event happened. It is written P(A|B) and equals P(A and B) divided by P(B).
It updates a probability when new evidence arrives: P(A|B) = P(B|A) x P(A) / P(B). You start with a prior belief, P(A), and adjust it by how likely the evidence is if A is true or false. It is used in medical testing, spam filters and risk assessment.
Because the condition may be rare. If 1% of people have it and the test wrongly flags 5% of healthy people, false positives from the 99% outnumber true positives from the 1%. With a 90% detection rate, a positive result means about a 15% chance. The Bayesian tab starts with exactly this example.
For independent events, multiply the probabilities: two heads in two coin flips is 0.5 x 0.5 = 0.25. For dependent events, multiply the first probability by the chance of the second given the first. Drawing two aces without putting the first back is 4/52 x 3/51, about 0.0045.
Add the two probabilities and subtract the chance of both: P(A or B) = P(A) + P(B) - P(A and B). The subtraction stops outcomes being counted twice. If the events cannot happen together, such as rolling a 1 or a 2 on one die, it is just 1/6 + 1/6 = 1/3.
For odds in favor, divide the chance it happens by the chance it does not: p / (1 - p). A probability of 0.75 gives 0.75 / 0.25, which is odds of 3 : 1. To go back, odds of a : b mean a probability of a / (a + b), so 3 : 1 is 3/4.

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