Probability Calculator
5 calculation modes
Number of outcomes you want
Total outcomes in the sample space
Probability per single trial (0 to 1)
Binomial probability: exactly k successes in n trials
nPr = n! / (n−r)! order matters
nCr = n! / (r! × (n−r)!): order does NOT matter
Quick Examples
Coin flip
P(heads) = 1/2
Card from deck
P(ace) = 4/52
Dice: even number
P(2,4,6) = 2/6
Poker hand
C(52,5) combinations
Choose a mode and enter values to calculate probability.
Probability
Formula
Step-by-Step
Probability Fundamentals
Probability is the number of favorable outcomes divided by the number of possible outcomes, a value from 0 (impossible) to 1 (certain): the chance of rolling a 6 on one die is 1/6, or 16.67%. This calculator handles single events, AND/OR for independent, dependent and mutually exclusive events, repeated trials, and permutations and combinations.
Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). P(event) = favorable outcomes / total outcomes. Multiply by 100 to get a percentage.
Single Event
P(A) = favorable / total. Example: rolling a 3 on a die = 1/6 ≈ 16.7%. Complement: P(not A) = 1 − P(A). If P(rain) = 0.3, P(no rain) = 0.7.
AND Probability
For independent events: P(A and B) = P(A) × P(B). Flipping heads twice: 0.5 × 0.5 = 0.25. For mutually exclusive: P(A and B) = 0 (cannot both occur).
OR Probability
P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events (like rolling 1 or 2): P(A or B) = P(A) + P(B) = 1/6 + 1/6 = 1/3.
Combinations vs Permutations
Combinations (nCr): order does not matter. Permutations (nPr): order matters. nPr = nCr × r!. Choosing 3 from 10: C(10,3) = 120 ways, P(10,3) = 720 ways.
Independent vs Dependent Events: Worked Examples
Independent: the first result does not change the second
A die has no memory, so the chance of three 6s in a row is 1/6 × 1/6 × 1/6 = 1/216, about 0.46%. Two heads in two coin flips is 0.5 × 0.5 = 0.25. Choose “Independent events” and the calculator multiplies P(A) by P(B).
Dependent: the first result changes what is left
Drawing two aces from a deck without putting the first card back: the first draw is 4/52, but if it was an ace only 3 aces remain among 51 cards. So P(both aces) = 4/52 × 3/51 = 12/2,652 = 1/221, about 0.45%. With replacement it would be 4/52 × 4/52 = 1/169, about 0.59%.
To run this in the calculator, choose “Dependent events”, enter P(A) = 4/52 = 0.0769 and, in the second box, the conditional probability P(B given A) = 3/51 = 0.0588. The rule is P(A and B) = P(A) × P(B given A). Three hearts in a row without replacement works the same way: 13/52 × 12/51 × 11/50 = 0.0129, or about 1 in 77.
Common Probabilities Reference Table
| Event | Probability | Percent | About 1 in |
|---|---|---|---|
| Heads on one coin flip | 1/2 | 50% | 2 |
| A 6 on one die | 1/6 | 16.67% | 6 |
| A total of 7 with two dice | 6/36 = 1/6 | 16.67% | 6 |
| Drawing a heart | 13/52 = 1/4 | 25% | 4 |
| Drawing an ace | 4/52 = 1/13 | 7.69% | 13 |
| At least one 6 in four rolls | 1 − (5/6)4 | 51.77% | 1.9 |
| Two of 23 people share a birthday | 1 − (365 × 364 × … × 343) ÷ 36523 | 50.73% | 2 |
| Three 6s in a row | 1/216 | 0.46% | 216 |
| Two aces in a row, no replacement | 1/221 | 0.45% | 221 |
| Royal flush in a 5-card hand | 4/2,598,960 | 0.00015% | 649,740 |
| Powerball jackpot, one ticket | 1/292,201,338 | 0.00000034% | 292,201,338 |
The birthday line ignores leap years and assumes birthdays are spread evenly, the usual textbook simplification.
Mistakes That Give Impossible Answers
- Typing a percentage as a whole number. Probabilities run from 0 to 1, so 30% is 0.3. The calculator rejects anything outside 0 to 1, because a probability above 1 or below 0 has no meaning.
- Adding events that can both happen. The chance of heads on the first or second flip is not 0.5 + 0.5 = 1. Subtract the overlap: 0.5 + 0.5 − 0.25 = 0.75. Only mutually exclusive events can simply be added.
- Treating dependent events as independent. Cards dealt from one deck, people picked from one group and parts pulled from one batch all change the odds for the next pick. Multiply by the conditional probability instead.
- Confusing mutually exclusive with independent. If A and B cannot happen together, knowing A happened tells you B did not, so they are dependent.
- The gambler’s fallacy. After five heads in a row, the next flip is still 50/50. Past independent results do not change the next one.
- Counting “at least one” directly. Use the complement: P(at least one) = 1 − P(none). At least one 6 in four rolls is 1 − (5/6)4 = 51.77%.
To turn a probability into a percentage or compare two chances, the percentage calculator and fraction calculator help.
Method and sources. Single events: favorable ÷ total. Independent events: P(A and B) = P(A) × P(B); dependent events: P(A) × P(B given A); P(A or B) = P(A) + P(B) − P(A and B). Repeated trials use the binomial formula C(n,k) pk(1 − p)n − k. Permutations and combinations are computed with the multiplicative formula so large n stays exact. Powerball odds from the game matrix (5 of 69 white balls and 1 of 26 Powerballs). Every value in the tables and examples above was computed and checked.
Probability Questions
Probability measures how likely an event is to occur. It is always between 0 (impossible) and 1 (certain). Formula: P(event) = number of favorable outcomes / total possible outcomes. Example: probability of rolling a 4 on a standard die = 1/6 ≈ 0.1667 ≈ 16.67%. Multiply by 100 to express as a percentage. The sum of probabilities of all possible outcomes in a sample space always equals exactly 1.
The complement of event A is "A does not occur," written as A' or Aᶜ. P(A') = 1 − P(A). If the probability of rain tomorrow is 0.3 (30%), then the probability of no rain = 1 − 0.3 = 0.7 (70%). Complements are useful because sometimes it is easier to calculate the probability of an event NOT occurring and subtract from 1. Example: probability of getting at least one head in 3 flips = 1 − P(no heads) = 1 − (0.5)³ = 1 − 0.125 = 0.875.
AND probability (intersection): both events occur. For independent events: P(A and B) = P(A) × P(B). Example: flipping heads AND rolling a 6 = 0.5 × 1/6 ≈ 0.083. OR probability (union): at least one event occurs. P(A or B) = P(A) + P(B) − P(A and B). Example: drawing a heart OR a face card from a deck = 13/52 + 12/52 − 3/52 = 22/52 ≈ 42.3%. The subtraction avoids double-counting outcomes where both occur.
Independent events: the outcome of one does not affect the other. Example: flipping a coin and rolling a die are independent. P(A and B) = P(A) × P(B). Mutually exclusive events: both cannot occur at the same time. Example: rolling a 1 and rolling a 2 on one die are mutually exclusive. P(A and B) = 0, so P(A or B) = P(A) + P(B). Caution: mutually exclusive events are NOT independent (if A occurs, B cannot, so they are dependent on each other).
Binomial probability calculates the chance of getting exactly k successes in n independent trials, where each trial has probability p of success. Formula: P(X=k) = C(n,k) × pᵏ × (1−p)^(n−k). Example: probability of getting exactly 3 heads in 10 coin flips. P(X=3) = C(10,3) × 0.5³ × 0.5&sup7; = 120 × 0.125 × 0.0078125 ≈ 0.117 = 11.7%. The binomial distribution describes many real-world scenarios: defect rates, survey responses, medical test results.
Permutations (nPr): arrangements where order matters. nPr = n!/(n−r)!. Example: how many 3-digit PINs from digits 1 to 9 (no repeats)? 9P3 = 9!/6! = 504. Combinations (nCr): selections where order does not matter. nCr = n!/(r!(n−r)!). Example: choosing 3 people from 9 for a committee (the order of selection doesn't matter): C(9,3) = 84. Relationship: nPr = nCr × r! (permutations = combinations × the number of ways to arrange the chosen items).
Odds express the ratio of favorable to unfavorable outcomes. Odds in favor of A = P(A) / P(A') = favorable / unfavorable. Example: rolling a 6 has probability 1/6. Odds in favor = 1:5 (1 favorable, 5 unfavorable). Odds against = 5:1. Converting: if odds in favor = a:b, then P = a/(a+b). If odds = 3:1 in favor, P = 3/4 = 0.75. Probability and odds describe the same information differently. Odds are commonly used in gambling and sports betting, while probability is used in science and statistics.
Conditional probability P(A|B) is the probability that A occurs given that B has already occurred. Formula: P(A|B) = P(A and B) / P(B). Example: a bag has 3 red and 2 blue balls. Probability of drawing red on the second draw given red on the first (without replacement): P(red first) = 3/5. P(red first AND red second) = 3/5 × 2/4 = 6/20 = 3/10. P(red second | red first) = (3/10) / (3/5) = 1/2. Bayes' theorem uses conditional probability to update beliefs based on new evidence.
Insurance: companies calculate the probability of claims to set premiums. Medicine: clinical trials measure the probability that a treatment works vs placebo. Weather forecasting: "70% chance of rain" is a probability statement. Finance: options pricing (Black-Scholes) uses probability that an asset reaches a certain price. Quality control: manufacturers calculate defect probabilities to set acceptance thresholds. Gambling: casinos design games with specific house edge probabilities. Epidemiology: tracking disease spread probabilities to guide public health decisions. Machine learning: algorithms assign probabilities to classifications.
Lottery probability uses combinations. For a typical 6/49 lottery (pick 6 numbers from 1 to 49): total combinations = C(49,6) = 13,983,816. Probability of jackpot = 1/13,983,816 ≈ 0.0000000715 or about 1 in 14 million. For Powerball (choose 5 from 69, plus 1 Powerball from 26): C(69,5) × 26 = 11,238,513 × 26 = 292,201,338. Jackpot probability = 1 in ~292 million. Even buying one ticket every week for 50 years (2,600 tickets) gives only about a 1 in 112,000 chance of ever hitting the jackpot.
Multiply the probability of the first event by the probability of the second given that the first happened: P(A and B) = P(A) × P(B given A). A bag with 3 red and 2 blue balls: the chance of two reds without replacement is 3/5 × 2/4 = 6/20 = 0.3. With replacement it would be 3/5 × 3/5 = 0.36, because the second draw is back to 3 reds out of 5.
No. Every probability is between 0 and 1, or 0% and 100%. A result above 1 means something went wrong, usually adding the chances of events that can happen together without subtracting the overlap, or entering a percentage such as 30 instead of 0.3. A negative result usually comes from subtracting in the wrong order, for example a complement worked out as P(A) − 1 instead of 1 − P(A).