How to use the probability calculator
- Pick a calculation with the buttons: Single event, Two events, Conditional, Dice, Coin flips or Combinations.
- Fill in the boxes, such as Favorable outcomes and Total outcomes. Counts must be whole numbers. Chances such as P(A) can be typed as 0.25, 25% or 1/4.
- Read the answer as a percent, with the decimal, fraction, odds or "1 in N" underneath. Anything invalid is explained next to the box.
The formulas behind each tab
With equally likely outcomes, probability = favorable ÷ total, and the chance something does not happen is 1 minus that. Independent events follow P(A and B) = P(A) × P(B) and P(A or B) = P(A) + P(B) - P(A and B); mutually exclusive events have P(A and B) = 0. Conditional probability is P(A | B) = P(A and B) ÷ P(B). The dice tab counts every way to reach the total and divides by sidesdice. Coin flips use the binomial formula P(k) = C(n,k) × pk × (1 - p)n-k, where C(n,k) = n! ÷ (k! × (n-k)!). Definitions follow OpenStax College Algebra and the NIST/SEMATECH handbook.
Worked example with the starting numbers
Single event opens with 13 favorable outcomes out of 52 total, the hearts in a deck. 13 ÷ 52 = 0.25, so the tool shows 25%, the fraction 1/4, odds for of 1 to 3, odds against of 3 to 1, and a 75% chance it is not a heart. The other tabs open with examples too. P(A) = 0.5 and P(B) = 0.4 give 20% for both and 70% for either. Two six-sided dice summing to 7 give 6 of 36 ways, or 16.6667%. Five heads in 10 fair flips is 24.6094%. Choosing 3 of 10 items gives 120 combinations and 720 permutations.
Limits and edge cases
A chance must be from 0 to 1, so typing 50 is rejected with a hint to use 50% or 0.5 instead of being guessed. Tiny results are shown with extra digits or in scientific notation rather than as 0%, and results just under 100% are not rounded up to it. Dice allows up to 50 dice with up to 100 sides, coin flips up to 1,000 flips and combinations up to 1,000 items, all with exact whole-number arithmetic. The Two events tab assumes independence unless you choose Mutually exclusive. See the scientific notation converter for tiny answers, and the dice roller or coin flip tool to simulate the odds.
Probability FAQ
What is the probability of rolling a 7 with two dice?
It is 6 out of 36, or about 16.67%. Six of the 36 equally likely rolls add up to 7: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1.
How do I find the probability of at least one success?
Find the chance of none and subtract it from 1. For at least one head in 3 fair flips, the chance of none is 0.5 × 0.5 × 0.5 = 0.125, so the answer is 1 - 0.125 = 87.5%.
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other and can both happen, so P(A and B) = P(A) × P(B). Mutually exclusive events cannot both happen, so P(A and B) = 0.
How are odds different from probability?
Probability compares favorable outcomes with all outcomes, while odds compare favorable with unfavorable. A 25% chance is 1 in 4, which is odds of 1 to 3 for the event.
Does a streak of heads make tails more likely next?
No. Each fair flip is independent, so the chance of heads stays 50% after any streak. Ten heads in a row is rare, about 0.0977%, but that is about the whole run, not the next flip.
Sources: OpenStax, College Algebra 2e, Probability; OpenStax, Introductory Statistics 2e, Independent and Mutually Exclusive Events; NIST/SEMATECH e-Handbook, Binomial Distribution.