Math & Statistics
Probability, Combinations & Permutations Calculator – Solve Math Problems Online
Use our free Probability, Combinations & Permutations Calculator to calculate probabilities, factorials, permutations, and combinations easily. Perfect for students and math enthusiasts.
What is Probability?
Probability is a branch of mathematics that measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). The basic formula is: P(A) = Favorable Outcomes ÷ Total Outcomes. For example, the probability of rolling a 3 on a fair die is 1/6 ≈ 0.1667 or 16.67%.
This calculator handles four types of calculations: Basic Probability (simple event likelihood), Combinations (selections where order doesn't matter), Permutations (arrangements where order matters), and Binomial Probability (probability of exactly k successes in n independent trials).
Probability Formulas Reference
| Type | Formula | Description | Example |
|---|---|---|---|
| Basic Probability | P(A) = favorable ÷ total | Likelihood of a single event | Rolling a 4: P = 1/6 = 16.67% |
| Complement | P(A') = 1 − P(A) | Probability event does NOT occur | NOT rolling a 4: 1 − 1/6 = 83.33% |
| Combination C(n,r) | n! ÷ (r! × (n−r)!) | Selections where order doesn't matter | C(5,2) = 10 ways |
| Permutation P(n,r) | n! ÷ (n−r)! | Arrangements where order matters | P(5,2) = 20 ways |
| Binomial | C(n,k) × pk × (1−p)n−k | Exactly k successes in n trials | 3 heads in 5 flips: 31.25% |
| AND (independent) | P(A ∩ B) = P(A) × P(B) | Both events occur | Two heads: 0.5 × 0.5 = 25% |
| OR (mutually exclusive) | P(A ∪ B) = P(A) + P(B) | Either event occurs | Rolling 1 or 2: 1/6 + 1/6 = 33.33% |
Common Probability Examples
| Scenario | Probability | Fraction | Percentage |
|---|---|---|---|
| Coin flip — heads | 1 in 2 | 1/2 | 50% |
| Rolling a specific number on a die | 1 in 6 | 1/6 | 16.67% |
| Drawing an ace from a deck | 4 in 52 | 1/13 | 7.69% |
| Drawing a heart from a deck | 13 in 52 | 1/4 | 25% |
| Rolling doubles with two dice | 6 in 36 | 1/6 | 16.67% |
| Rolling a 7 with two dice | 6 in 36 | 1/6 | 16.67% |
| Birthday shared in a group of 23 | — | — | 50.73% |
| Lottery (6 from 49) | 1 in 13,983,816 | — | 0.0000072% |
Combinations vs Permutations — When to Use Which?
Combinations (Order Doesn't Matter)
Use when you're selecting a group and the arrangement doesn't matter. Choosing 3 people from 10 for a committee — {A,B,C} is the same as {C,A,B}.
- Lottery numbers
- Choosing team members
- Selecting menu items
- Card hands (poker, etc.)
Permutations (Order Matters)
Use when the arrangement or sequence matters. Choosing 3 people from 10 for president, VP, and secretary — {A,B,C} is different from {C,A,B}.
- PIN codes and passwords
- Race finishing positions
- Seating arrangements
- Phone numbers
FAQ – Probability Calculator
What is probability and how is it calculated?
Probability measures the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). The basic formula is P(A) = Number of favorable outcomes ÷ Total number of possible outcomes. For example, the probability of drawing a king from a standard deck is 4/52 = 1/13 ≈ 7.69%.
What is the difference between combinations and permutations?
Combinations count selections where order doesn't matter — choosing 3 books from 10 gives C(10,3) = 120 groups. Permutations count arrangements where order matters — arranging 3 books from 10 on a shelf gives P(10,3) = 720 arrangements. Permutations are always ≥ combinations for the same n and r.
What is binomial probability?
Binomial probability calculates the chance of getting exactly k successes in n independent trials, where each trial has the same probability p of success. Formula: P(X=k) = C(n,k) × pk × (1−p)n−k. Example: probability of getting exactly 3 heads in 5 coin flips = C(5,3) × 0.5³ × 0.5² = 31.25%.
What is a factorial and how is it used in probability?
A factorial (n!) is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1. Factorials are used in combination and permutation formulas to count the number of ways to arrange or select items from a set.
What is the complement of a probability?
The complement P(A') is the probability that event A does NOT occur: P(A') = 1 − P(A). This is useful when it's easier to calculate the probability of something not happening. For example, the probability of rolling at least one 6 in 4 rolls = 1 − (5/6)⁴ = 1 − 0.482 = 51.8%.
How do I calculate the probability of two events both happening?
For independent events (one doesn't affect the other), multiply their probabilities: P(A AND B) = P(A) × P(B). For example, flipping heads twice: 0.5 × 0.5 = 0.25 (25%). For dependent events, use conditional probability: P(A AND B) = P(A) × P(B|A).
How do I calculate the probability of either event happening?
For mutually exclusive events (can't happen together), add probabilities: P(A OR B) = P(A) + P(B). For non-mutually exclusive events, subtract the overlap: P(A OR B) = P(A) + P(B) − P(A AND B). Example: drawing a king or a heart = 4/52 + 13/52 − 1/52 = 16/52 ≈ 30.8%.
What is the birthday problem?
The birthday problem asks: how many people are needed for a 50% chance that two share a birthday? The surprising answer is just 23. With 70 people, the probability exceeds 99.9%. This counterintuitive result occurs because we're comparing all possible pairs, not just one person against the rest.
What are independent vs dependent events?
Independent events don't affect each other — flipping a coin twice, rolling dice. The outcome of one has no effect on the other. Dependent events are linked — drawing cards without replacement. After drawing one card, the remaining deck changes, altering probabilities for the next draw.
Can probability be greater than 1 or less than 0?
No. Probability always ranges from 0 to 1 (or 0% to 100%). A probability of 0 means the event is impossible, and 1 means it's certain. If your calculation gives a value outside this range, there's an error in the inputs or formula. Combinations and permutations can be large numbers, but they represent counts, not probabilities.