Probability Calculator
Calculate probability as a fraction, decimal, percentage, and odds. Enter favorable and total outcomes for single-event probability, or use the compound section for P(A and B), P(A or B), and conditional probability.
Last updated: March 2026
Single Event Probability
Compound Probability
How to Use the Probability Calculator
- Single event: Enter the number of favorable outcomes and total outcomes. Probability, decimal, percentage, and odds appear instantly.
- Compound probability: Enter P(A) and P(B) between 0 and 1, then select the event relationship (independent or mutually exclusive).
- Read all results — P(A and B), P(A or B), and the complement P(not A) are shown simultaneously.
- Share or clear using the buttons below the form.
Understanding Probability: The Basics
Probability measures the likelihood that an event will occur, expressed as a number between 0 (impossible) and 1 (certain). A probability of 0.5 means the event is equally likely to occur or not occur. There are three ways to express probability:
- Fraction: 1/6 (1 favorable out of 6 possible outcomes)
- Decimal: 0.1667
- Percentage: 16.67%
Classical, Empirical, and Subjective Probability
Classical probability is based on equally likely outcomes (theoretical): rolling a die, flipping a coin, drawing cards. Empirical probability is based on observed frequencies: if it rained 60 out of 100 similar days, P(rain) = 0.6. Subjective probability is based on judgment and experience: a weather forecaster's 70% chance of rain, or a doctor's assessment of a diagnosis.
Odds vs. Probability
Odds express the ratio of favorable outcomes to unfavorable outcomes, while probability is the ratio of favorable to total. For P = 1/4: there are 1 favorable and 3 unfavorable, so odds are 1:3 (read as "1 to 3"). To convert: P = odds_for / (odds_for + odds_against). Sports betting uses odds extensively — understanding the difference between probability and odds is essential for interpreting betting lines.
Compound Probability: Rules and Formulas
When dealing with two or more events, the way you combine probabilities depends on the relationship between the events.
Multiplication Rule (AND)
For independent events: P(A and B) = P(A) × P(B). The outcome of A doesn't affect B. Example: P(heads) × P(rolling a 6) = 0.5 × (1/6) ≈ 0.083.
For dependent events: P(A and B) = P(A) × P(B|A), where P(B|A) is the conditional probability of B given A. Drawing two aces without replacement: P(1st ace) = 4/52, P(2nd ace | 1st ace drawn) = 3/51. P(both aces) = (4/52) × (3/51) ≈ 0.0045.
Addition Rule (OR)
For mutually exclusive events: P(A or B) = P(A) + P(B). Rolling a 2 or a 5 on a die: 1/6 + 1/6 = 2/6 = 1/3.
For non-mutually exclusive events: P(A or B) = P(A) + P(B) − P(A and B). Drawing a heart or a queen: P(heart) = 13/52, P(queen) = 4/52, P(queen of hearts) = 1/52. P(heart or queen) = 13/52 + 4/52 − 1/52 = 16/52 ≈ 0.308.
Complement Rule
P(not A) = 1 − P(A). This is one of the most useful shortcuts in probability. Instead of calculating P(at least one head in 5 flips) directly, calculate 1 − P(no heads) = 1 − (0.5)⁵ = 1 − 0.03125 = 0.96875 = 96.875%.
Real-World Applications
- Cards: P(face card) = 12/52 = 3/13 ≈ 23.1%. P(red ace) = 2/52 = 1/26 ≈ 3.8%.
- Dice: P(sum = 7 with two dice) = 6/36 = 1/6 ≈ 16.7% (most likely sum).
- Weather: If P(rain on day 1) = 0.4 and P(rain on day 2) = 0.3 (independent), P(rain both days) = 0.12 = 12%.
- Quality control: If a machine produces 2% defective parts, P(both of 2 random parts are defective) = 0.02² = 0.04%.
How to Calculate Probability If...
What is the probability of rolling a 6 on a standard die?
P(rolling a 6) = 1/6 ≈ 0.1667 or 16.67%. There is 1 favorable outcome (the face showing 6) out of 6 equally likely outcomes. The odds are 1:5 (1 for, 5 against).
What is the probability of flipping heads three times in a row?
P(HHH) = (1/2)³ = 1/8 = 0.125 or 12.5%. Each flip is independent with P(heads) = 0.5. Multiply: 0.5 × 0.5 × 0.5 = 0.125.
What is the probability of drawing a red card from a standard deck?
P(red card) = 26/52 = 1/2 = 0.5 or 50%. There are 26 red cards (13 hearts + 13 diamonds) in a 52-card deck.
What is the probability of rolling an even number on a die?
P(even) = 3/6 = 1/2 = 0.5 or 50%. The even outcomes are {2, 4, 6} — 3 out of 6 possible outcomes.
What is the probability of getting at least one tail in two coin flips?
P(at least one tail) = 1 − P(no tails) = 1 − (1/2)² = 1 − 0.25 = 0.75 or 75%. Using the complement rule is faster than listing favorable outcomes (HT, TH, TT).
What is the probability of picking a vowel from the English alphabet?
P(vowel) = 5/26 ≈ 0.1923 or 19.23%. There are 5 vowels (A, E, I, O, U) in the 26-letter alphabet.
What is the probability that two independent events both fail?
P(both fail) = P(not A) × P(not B) = (1 − P(A)) × (1 − P(B)). If P(A) = 0.8 and P(B) = 0.9, then P(both fail) = 0.2 × 0.1 = 0.02 = 2%.
What is the probability of rolling a sum of 7 with two dice?
P(sum = 7) = 6/36 = 1/6 ≈ 16.67%. The pairs that sum to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 favorable outcomes out of 36 total.
What is the probability of a 1-in-365 event happening at least once in a year?
P(at least once in 365 tries) = 1 − (364/365)³⁶⁵ ≈ 63.2%. Even a rare daily event (0.27% chance) becomes more likely than not over a full year — a counterintuitive but important result.
What is the probability if favorable outcomes equal total outcomes?
If favorable = total, probability = 1 (certain event). For example, P(rolling any number 1–6 on a die) = 6/6 = 1. A probability of 1 means the event is guaranteed to occur.
People Also Search For
Probability calculator for dice and card games
For a single die: favorable outcomes = 1 (for a specific face), total = 6. For a deck of cards: total = 52, favorable = count of desired cards. Rolling a specific number: 1/6. Drawing a specific suit: 13/52 = 1/4. Drawing a specific card: 1/52. This calculator handles any combination of favorable and total outcomes.
How to simplify a probability as a fraction
Divide both numerator and denominator by their greatest common divisor (GCD). P = 12/52: GCD(12, 52) = 4, so 12/52 = 3/13. This calculator automatically shows the simplified fraction. A fraction in lowest terms has GCD(numerator, denominator) = 1.
Theoretical vs experimental probability
Theoretical probability is calculated from the structure of the experiment (equally likely outcomes); experimental probability is measured from actual trial results. Theoretical P(heads) = 0.5, but after 10 flips you might see heads 7 times (experimental P = 0.7). As trials increase, experimental probability converges to theoretical probability (the law of large numbers).
How to calculate P(A or B) for non-mutually exclusive events
Use the inclusion-exclusion formula: P(A or B) = P(A) + P(B) − P(A and B). Without subtracting P(A and B), you count the overlap twice. Drawing a king or a spade: P(king) = 4/52, P(spade) = 13/52, P(king of spades) = 1/52. P(king or spade) = 4/52 + 13/52 − 1/52 = 16/52 ≈ 30.8%.
Probability of multiple independent events all occurring
Multiply all individual probabilities: P(A and B and C...) = P(A) × P(B) × P(C) × ... P(rolling three 6s in a row) = (1/6)³ = 1/216 ≈ 0.46%. The probability of multiple independent successes decreases rapidly with each additional event.
Converting between probability and odds (betting)
To convert probability to odds: odds = P / (1 − P). To convert odds back: P = odds / (1 + odds). P = 0.25 → odds = 0.25/0.75 = 1/3 = "1 to 3". Bookmaker odds of 3/1 (3 against, 1 for) imply P = 1/(1+3) = 0.25. Understanding this conversion is essential for evaluating whether betting odds offer value.
Frequently Asked Questions
What is the probability of rolling a 6 on a die?
P(6) = 1/6 ≈ 16.67%. Enter 1 favorable outcome and 6 total outcomes. Odds are 1:5 (1 for, 5 against).
What is the difference between P(A and B) and P(A or B)?
P(A and B) = both events occur simultaneously (intersection). P(A or B) = at least one event occurs (union). For independent events: P(A and B) = P(A) × P(B). For mutually exclusive events: P(A or B) = P(A) + P(B).
What is the complement rule?
P(not A) = 1 − P(A). If P(rain) = 0.3, then P(no rain) = 0.7. The complement is the probability that an event does NOT occur. Complements always sum to 1.
What are mutually exclusive events?
Events that cannot both occur at the same time. Rolling a 3 and rolling a 5 are mutually exclusive (can't get both on one roll). For mutually exclusive events, P(A and B) = 0 and P(A or B) = P(A) + P(B).
What does probability 0 or 1 mean?
Probability 0 means an impossible event (will never occur). Probability 1 means a certain event (will always occur). All probabilities are between 0 and 1 inclusive.
How do I calculate odds from probability?
Odds for = favorable : unfavorable = P : (1 − P). If P = 0.25, odds = 0.25 : 0.75 = 1 : 3. This calculator automatically shows odds alongside probability.
What is conditional probability?
P(A|B) = probability of A given B has occurred = P(A and B) / P(B). Example: given a card drawn is a face card (B), P(it is a king | face card) = P(king and face) / P(face) = (4/52) / (12/52) = 4/12 = 1/3.
What is the probability of getting exactly 2 heads in 3 coin flips?
P = C(3,2) × (0.5)² × (0.5)¹ = 3 × 0.25 × 0.5 = 0.375 or 37.5%. The favorable outcomes are HHT, HTH, THH — 3 out of 8 total equally likely outcomes.
Is my data private?
Yes. All calculations run locally in your browser. No values are sent to any server or stored anywhere.
What is expected value?
Expected value (EV) = Σ [value × probability]. It is the long-run average outcome of a repeated experiment. A positive EV means you gain on average; a negative EV means you lose. Casino games are designed to have negative EV for the player.