Odds from outcomes, or permutations and combinations.
Probability
—
—
Decimal
—
Odds
How this works
Probability = favourable outcomes ÷ total outcomes
nPr (order matters) = n! ÷ (n − r)!
nCr (order does not matter) = n! ÷ (r! × (n − r)!)
Probability always sits between 0 (impossible) and 1 (certain) — as a percent, between 0% and 100%.
Odds "a to b" mean a favourable outcomes for every b unfavourable ones — different from probability, which compares favourable to total.
Use permutations when order matters (arranging people in a line); use combinations when it does not (choosing a committee).
What is the difference between probability and odds?
Probability compares favourable outcomes to all possible outcomes (3 in 6 = 0.5). Odds compare favourable outcomes to unfavourable ones (3 to 3, or 1 to 1) — the same situation, expressed two different ways.
When do I use a permutation instead of a combination?
Use a permutation (nPr) when the order of selection matters, like ranking 1st, 2nd and 3rd place. Use a combination (nCr) when only the group chosen matters, like picking 3 people for a team, with no ranking.
Why is nCr always smaller than nPr for the same n and r?
nPr counts every ordering of the same r items as a separate outcome. nCr groups all those orderings together as one outcome, so it divides nPr by r! (the number of ways to order r items).