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Combinatorial Probability Calculator

Probability from favorable and total outcomes

How it is calculated

The tool applies classical probability: P = f / t, where f is the number of favorable outcomes and t the total number of equally likely outcomes. The probability is shown with two decimals, or with more when it is below 0.01 (6 of 1326 gives 0.00452), so small values do not collapse to 0.00. The percentage is P × 100. The odds are shown as odds against: (t − f) / f : 1, meaning how many unfavorable outcomes there are for each favorable one. Both counts are whole numbers; any decimals you type are cut off (2.9 is read as 2). The result is empty if the total is zero, if the favorable count is negative, or if it is larger than the total. With zero favorable outcomes the odds read "never". The formula is only valid when every outcome is equally likely.

Worked example

What is the chance of drawing an ace from a standard 52-card deck? There are 4 aces, so f = 4 and t = 52. Probability: 4 / 52 = 0.08 (rounded to two decimals; the exact value is 0.0769...). As a percentage this is 7.69%. The unfavorable outcomes number 52 − 4 = 48, so the odds against are 48 / 4 = 12.00 : 1. In other words, you expect about 12 non-aces for every ace.

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Common mistakes

  • Reading the odds as favorable : unfavorable. The tool gives odds against (unfavorable per one favorable), so 12.00 : 1 means 12 misses for each hit, not the other way around.
  • Using total outcomes that are not equally likely, such as the sum of two dice (7 is more likely than 2). Count the 36 ordered rolls, not the 11 possible sums.
  • Entering the probability as a percentage or fraction instead of counts. The fields expect whole numbers of outcomes.
  • Typing a decimal count such as 2.5, which is silently cut to 2 and changes the result.
  • Forgetting that order matters or does not matter when you work out the counts with permutations or combinations first, so the favorable and total counts are not counted the same way.

About this calculator

This calculator computes classical probability as favorable outcomes ÷ total equally-likely outcomes, and converts the same figure into odds against (unfavorable : favorable, shown as a ratio to 1) for anyone more used to reading betting-style ratios. It's the definition behind every textbook dice, card and coin problem — count what counts as a "win," count everything that can happen, and divide.

Students use it for combinatorics homework where the favorable and total counts come from permutations or combinations worked out beforehand, and it's the same arithmetic bettors and game designers use to convert a raw outcome count into a probability or a payout ratio.

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Comments (1)

  • Editorial team

    The tool only divides the two counts you give it, so the real work is counting both with the same rules. For two aces dealt from a 52-card deck, favourable is 6 pairs and total is 1326 pairs, not 4 and 52. For rare events the first line keeps extra decimals, here 0.00452, and the percent line reads 0.45 percent, with odds against of 220 to 1. Which situation are you trying to count?

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