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Probability calculator

A count of favourable outcomes and a count of all of them, or a list of independent probabilities, turned into a chance, a complement and odds.

Single event with equally likely outcomes

Independent events (leave empty for none)

Short answer: rolling a 6 on a fair die is P = 0.1667, so 0.8333 it does not happen and odds of 1 : 5. Two coins both showing heads is 0.25, and at least one head is 0.75.

Counting versus measuring

Probability from counts is exact and needs the assumption of equally likely outcomes. Probability from data — how often a bus is late — is measured, and the two must not be mixed: a bus that is late 3 days in 10 has a frequency of 0.3, not a probability of three tenths in any exact sense.

Worked example

A bag holds 4 red and 6 blue balls. Drawing red is 4 ÷ 10 = 0.4 and blue 0.6, with odds of 4 : 6. Drawing two reds without replacing the first is 0.4 × 3/9 = 0.1333 — and that second factor is why the page insists the independence assumption be checked before multiplying.

Probability questions

What does “equally likely” mean here?

It means every outcome has the same chance, as with a fair die, a shuffled deck or a bag you cannot see into. The count method only works under that assumption — weather and most real-world events are not equally likely and are not this page’s job.

How do I combine several events?

Multiply for “all of them happen”: two coins both landing heads is ½ × ½ = 0.25. For “at least one”, work out the chance that none happen and subtract it from 1, which is 0.75 for the same two coins.

What are odds and how do they differ from probability?

Probability is favourable over total; odds are favourable against unfavourable. A 1-in-6 chance is odds of 1 : 5 for and 5 : 1 against, and bookmakers use the second form, which is why a “5 to 1” payout looks generous next to a 17% chance.

Are the events required to be independent?

For the combined section, yes. A coin does not remember the last flip, but rain today and rain tomorrow are related, and drawing cards without replacing them changes the deck. The page states the assumption instead of quietly applying it.

Related tools

Percentage calculatorPercentages and change.Statistics calculatorMean, median and spread.Permutation and combination calculatorCounting outcomes without listing them.