Probability Calculator
The probability calculator covers three common probability calculations: basic event probability, combinations nCr, and permutations nPr. Basic probability compares favorable outcomes with total possible outcomes. Combinations count selections where order does not matter, while permutations count arrangements where order matters. Each mode shows the formula used and the worked calculation so you learn alongside getting the answer for coins, dice, cards, passwords, contests, and event planning.
Probability Visual
How to Use
- Select Basic Probability when you know the number of successful outcomes and the total possible outcomes.
- Select Combinations when you are choosing items and order does not matter, such as choosing 3 students from a group of 10.
- Select Permutations when order matters, such as arranging prize winners as first, second, and third place.
- Enter non-negative whole numbers. For nCr and nPr, n must be greater than or equal to r.
- Click Calculate to see the formula, exact count, decimal probability, or percentage depending on the selected mode.
Probability Formulas
nCr = n! / (r!(n-r)!)
nPr = n! / (n-r)!
Basic probability measures the chance that a specific event happens. Favorable outcomes are the outcomes you want, and total outcomes are all equally likely possibilities. If one die has six faces and only one face is a six, the probability of rolling a six is 1/6. Combinations count selections where order does not matter. Choosing A, B, C is the same as choosing C, B, A, so duplicate orders are divided out using r!. Permutations count arrangements where order matters. A first-place, second-place, and third-place finish is different from the same people in another order, so fewer duplicates are removed. Factorial notation n! means multiplying every positive integer from n down to 1. These formulas are fundamental in probability, statistics, lotteries, cards, scheduling, and counting problems.
Worked Example
Suppose a club has 10 members and needs to choose 3 people for a committee. Order does not matter because a committee with A, B, and C is the same as one with C, A, and B. Use combinations: 10C3 = 10! / (3! × 7!). Cancel 7! from the numerator and denominator, leaving (10 × 9 × 8) / (3 × 2 × 1). The numerator is 720 and the denominator is 6, so 10C3 = 120. If instead the club assigned president, secretary, and treasurer, order would matter. Then 10P3 = 10 × 9 × 8 = 720 possible assignments.
Interpreting Probability Results
A probability result is easiest to understand when shown in several forms. A fraction such as 1/6 is exact and useful for theoretical work. A decimal such as 0.1667 is useful for computation. A percentage such as 16.67% is easiest to communicate in everyday language. For combinations and permutations, the result is a count of possible outcomes, not a probability by itself. To turn a count into probability, compare favorable arrangements with total arrangements. For example, if 6 arrangements out of 36 possible dice outcomes meet your condition, the probability is 6/36 = 1/6.
Common Probability Examples
| Event | Favorable | Total | Probability |
|---|---|---|---|
| Coin heads | 1 | 2 | 50% |
| Dice rolls 6 | 1 | 6 | 16.67% |
| Dice rolls even | 3 | 6 | 50% |
| Card is Ace | 4 | 52 | 7.69% |
| Card is red | 26 | 52 | 50% |
| Two heads (2 coins) | 1 | 4 | 25% |
| Sum 7 on two dice | 6 | 36 | 16.67% |
| Birth on weekend | 2 | 7 | 28.57% |
Probability FAQ
What is probability
Probability is a number that describes how likely an event is to happen. It ranges from 0 to 1, or from 0% to 100%. A probability of 0 means the event cannot happen, while a probability of 1 means it is certain. Most real events fall between those extremes. In simple equally likely cases, probability is favorable outcomes divided by total outcomes. For example, a fair coin has one heads side and two possible sides, so the probability of heads is 1/2 or 50%.
What is the difference between combinations and permutations
Combinations count selections where order does not matter. If you choose three books to take on a trip, the same three books count once no matter which one you name first. Permutations count arrangements where order matters. If three runners win gold, silver, and bronze, changing the order changes the result. This difference is why nCr divides by r! to remove duplicate orders, while nPr does not. Ask whether rearranging the selected items creates a new outcome; if yes, use permutations.
What does nCr mean
nCr means the number of ways to choose r items from n items when order does not matter. The “n” is the total number of available items, and “r” is the number chosen. For example, 5C2 counts pairs chosen from five people. The formula is n! divided by r! times (n-r)!. It removes duplicate arrangements because choosing Ana then Ravi is the same pair as choosing Ravi then Ana. Combinations are used in committees, lottery tickets, card hands, and sample selection.
What does nPr mean
nPr means the number of ways to arrange r items chosen from n items when order matters. The first position has n choices, the second has n-1 choices, and so on until r positions are filled. For example, 5P2 = 5 × 4 = 20 because there are five choices for first place and four remaining choices for second place. Permutations are used for rankings, passwords, race results, seating arrangements, and ordered schedules where position changes the outcome.
Can probability be greater than 100%
No. A valid probability cannot be less than 0 or greater than 1, which is the same as 0% to 100%. If a calculation gives more than 100%, the inputs or assumptions are wrong. Common mistakes include counting overlapping outcomes twice, using a favorable count larger than the total count, or applying a formula where outcomes are not equally likely. In counting problems, nCr and nPr give numbers of arrangements, not probabilities by themselves. To turn them into probability, divide favorable arrangements by total arrangements.