Probability Calculator
Calculate probability, combinations (nCr), permutations (nPr) and independent events with formulas and step-by-step working
Probability
0.5000
50.00% chance
Formula
P(Event) = Favorable Outcomes / Total Outcomes
P = 3 / 6 = 0.5000
What the Probability Calculator Covers
Four tabs handle four core questions. Basic computes the probability of a single event as favourable outcomes divided by total outcomes. Combinations (nCr) tells you how many ways you can choose r items from n when order doesn't matter. Permutations (nPr) does the same when order matters. Independent events combines two probabilities to find the chance of both happening, either happening, or neither happening.
Probability values always sit between 0 (impossible) and 1 (certain), and the calculator shows them both as a decimal and a percentage. A probability of 0.25 is the same as a 25% chance, or odds of 1 in 4. The calculator doesn't deal with conditional probability or Bayesian updating directly, those are deeper topics best handled with a worked example rather than a generic tool.
Combinations vs Permutations
The classic way to remember the difference: combinations are about committees, permutations are about podiums. If you're picking 3 people from 10 to form a committee, the order they're picked in doesn't matter, so use nCr. If you're picking 3 people from 10 to come 1st, 2nd and 3rd in a race, the order matters (1st place is different from 3rd), so use nPr. The formulas: nCr = n! / (r! × (n-r)!), nPr = n! / (n-r)!. Permutations are always at least as large as combinations because they count more arrangements as distinct.
Common surprises: the number of possible bridge hands (13 cards from 52) is C(52, 13) = 635 billion. The number of distinct handshakes in a room of 20 people is C(20, 2) = 190. The number of ways to order a deck of 52 cards is 52! which is roughly 8 × 10^67, more than the number of atoms in the Earth.
Common Probability Examples
| Event | Probability | Approximate odds |
|---|---|---|
| Flipping heads | 0.5 (50%) | 1 in 2 |
| Rolling a 6 on a die | 0.167 (16.7%) | 1 in 6 |
| Drawing an ace from a deck | 0.077 (7.7%) | 1 in 13 |
| Two coins both heads | 0.25 (25%) | 1 in 4 |
| Three coins all heads | 0.125 (12.5%) | 1 in 8 |
| Rolling a double 6 with two dice | 0.028 (2.8%) | 1 in 36 |
| UK Lotto jackpot (6 from 59) | 0.0000000223 (0.00000223%) | 1 in 45,057,474 |
How Independent Events Combine
Two events A and B are independent if the outcome of one doesn't affect the other. For independent events: P(A and B) = P(A) × P(B), P(A or B) = P(A) + P(B) - P(A) × P(B). Multiplying probabilities lowers them quickly, which is why getting four heads in a row drops to (1/2)^4 = 1/16 even though each flip is 50/50. The 'or' formula subtracts the overlap because P(A) + P(B) on its own would double-count the case where both happen.
Common mistakes: assuming independence when the events are actually linked (drawing two cards without replacement is not independent because removing the first card changes the deck), or forgetting the subtraction in the 'or' formula. For dependent events you'd need conditional probability, which the [Standard Deviation Calculator](/standard-deviation-calculator) doesn't cover but is closely related to via Bayes' theorem.
Frequently Asked Questions
What's the difference between a combination and a permutation?
Order. Picking 3 people for a committee from 10 is C(10, 3) = 120 ways because no one cares who was picked first. Picking 3 people for gold, silver and bronze medals from 10 athletes is P(10, 3) = 720 ways because the medal order is part of the answer. P(n, r) = C(n, r) × r! always, since for each unordered combination there are r! ways to arrange it.
Why does the calculator show probabilities in two formats?
Decimals between 0 and 1 are the mathematical standard and the form used in formulas, while percentages between 0% and 100% are how most people speak about chance. A 0.045 probability and a 4.5% chance are exactly the same thing, just different notations. Odds (like '1 in 22') are a third notation, useful for headline numbers but harder to combine algebraically.
What happens when r is greater than n?
C(n, r) and P(n, r) are both zero. You can't pick more items than exist. The calculator returns 0 in that case, which can be confusing if you've made a typo, since the result looks the same as a genuine zero probability. Double-check that n is the larger of the two values.
Are coin flips and dice rolls really independent?
Yes, in any practical sense. The result of one fair coin flip has no physical influence on the next. The 'gambler's fallacy' is the false belief that, after a streak of heads, tails is somehow 'due'; the coin doesn't know what it did before, and each flip is 50/50 fresh. The same applies to dice and lottery balls.
How do I calculate the probability of getting at least one success?
Use the complement: P(at least one) = 1 - P(none). For example, the probability of rolling at least one 6 in four rolls is 1 - (5/6)^4 = 1 - 0.482 = 0.518, just over half. This 'at least one' technique is much easier than enumerating all the cases where you get one, two, three or four sixes.