- Single Event. Enter the number of favorable outcomes and total possible outcomes. P(A) = favorable / total. Rolling a 3 on a die: 1 favorable out of 6 total = 0.1667 (16.67%). The result also shows P(not A), odds in favor, and odds against.
- Two Events. Enter P(A) and P(B) as decimals between 0 and 1. Check the "independent" box if the events do not affect each other (like two coin flips). Results include P(A and B), P(A or B), P(A given B), and P(neither A nor B).
- Combination C(n,r). Enter n (total items) and r (items chosen). Order does not matter. Example: choosing 3 people from a group of 10 gives C(10,3) = 120 possible groups.
- Permutation P(n,r). Enter n and r when order matters. Arranging 3 people from 10 in specific positions gives P(10,3) = 720 arrangements.
Probability Calculator
Calculate the probability of events, combinations, and permutations.
Probability
0.5000
50.00%
| P(not A) | 0.5000 (50.00%) |
| Odds in favor | 3:3 |
| Odds against | 3:3 |
How to Use the Probability Calculator
Probability Formulas
Single Event: P(A) = favorable outcomes / total outcomes Complement: P(not A) = 1 - P(A) Independent AND: P(A and B) = P(A) x P(B) Dependent AND: P(A and B) = P(A) x P(B|A) OR (any events): P(A or B) = P(A) + P(B) - P(A and B) Mutually exclusive: P(A or B) = P(A) + P(B) Combination C(n,r): n! / (r! x (n-r)!) Permutation P(n,r): n! / (n-r)!
Examples with a standard deck of 52 cards:
| Question | Calculation | Result |
|---|---|---|
| Drawing an ace | 4/52 | 7.69% |
| Drawing a heart | 13/52 | 25% |
| Ace or heart | 4/52 + 13/52 - 1/52 | 30.77% |
| Two aces in a row (no replace) | 4/52 x 3/51 | 0.45% |
| 5-card hands (order irrelevant) | C(52,5) | 2,598,960 |
Frequently Asked Questions
Probability measures how likely an event is to occur, expressed as a number from 0 (impossible) to 1 (certain), or as a percentage from 0% to 100%. The basic formula is: P(event) = favorable outcomes / total possible outcomes. Rolling a 4 on a six-sided die: 1/6 = 0.167 = 16.7%. Drawing a red card from a standard deck: 26/52 = 0.5 = 50%.
A combination counts selections where order does not matter. Picking 3 toppings for a pizza from 10 options: C(10,3) = 120 possible combinations. A permutation counts arrangements where order matters. Assigning 1st, 2nd, 3rd place to 10 runners: P(10,3) = 720 arrangements. The relationship: C(n,r) = P(n,r) / r!, because combinations ignore the r! ways to order selected items.
Two events are independent if the outcome of one does not affect the probability of the other. Flipping a fair coin twice: the first flip (heads or tails) has zero effect on the second. Drawing cards with replacement is independent. Drawing without replacement is dependent because removing one card changes the deck composition for the next draw.
P(A and B) is the probability both events happen simultaneously (the intersection). For independent events, P(A and B) = P(A) x P(B). P(A or B) is the probability at least one event occurs (the union). The addition rule: P(A or B) = P(A) + P(B) - P(A and B). The subtraction prevents double-counting outcomes where both happen. For mutually exclusive events (can't both occur), P(A and B) = 0, so P(A or B) = P(A) + P(B).
Odds and probability express the same information differently. If P(A) = 1/4 (25%), the odds in favor are 1:3 (1 way to win, 3 ways to lose). Odds against are 3:1. To convert: probability = favorable / (favorable + unfavorable). Odds in favor = P(A) / P(not A). Sports betting uses odds to imply probability, though bookmakers build in a margin, so implied probabilities sum to more than 100%.