Probability Calculator
Calculate basic probability and combined events.
Single Event
Two Events (Independent)
Quick Examples
What Is Probability?
Probability is a way of thinking about chance: it quantifies how likely an event is on a scale from 0 (impossible) to 1 (certain). The classical formula counts outcomes — favorable outcomes divided by total outcomes — and it works whenever every outcome is equally likely, such as a fair die, a shuffled deck, or a random draw.
Not all probabilities come from counting. Probability can be theoretical (reasoned from symmetry, like 1/6 for a die face), experimental (estimated from observed data, like a bus being late 6 times out of 10), or subjective (a judgment call based on experience). The rules below apply to all three.
Probability Formulas
Key Rules
- Complement rule: P(not A) = 1 − P(A). Often the "not" event is easier to compute — for example, the chance of at least one success is 1 minus the chance of none.
- Addition rule (mutually exclusive): if A and B cannot happen together, P(A or B) = P(A) + P(B).
- Addition rule (general): if they can overlap, subtract the double-counted overlap: P(A or B) = P(A) + P(B) − P(A and B).
- Multiplication rule (independent): if one event does not affect the other, P(A and B) = P(A) × P(B).
- Multiplication rule (dependent): if it does, P(A and B) = P(A) × P(B|A).
Combined Events (Independent)
- P(A and B): P(A) × P(B)
- P(A or B): P(A) + P(B) - P(A and B)
Conditional Probability
P(A|B) is the probability of A given that B has already happened. Bayes' rule flips the condition: P(A|B) = P(B|A) × P(A) / P(B), which lets you update a probability when new evidence arrives — the engine behind spam filters and medical test interpretation.
Odds vs Probability
Probability compares favorable outcomes to all outcomes; odds compare them to the unfavorable ones. If the odds in favor are a : b, the probability is a / (a + b):
In the other direction, a probability p corresponds to odds of p : (1 − p) in favor. A probability of 0.25 is odds of 1 : 3, not 1 : 4.
Which Probability Question Do You Have?
| You Want | Use |
|---|---|
| The probability of one event from counts | P(A) = favorable / total |
| The chance that two events both happen | Multiplication rule (check independence) |
| The chance that either happens | Addition rule (subtract overlap if not exclusive) |
| The probability of A given B | Conditional probability / Bayes' rule |
| At least one success in n tries | 1 − P(none) via the complement rule |
| k successes in n independent tries | Binomial distribution |
| k successes drawing without replacement | Hypergeometric distribution |
Worked Examples
Example 1: Drawing a Heart
What is the probability of drawing a heart from a standard 52-card deck? There are 13 favorable outcomes out of 52: P = 13/52 = 1/4 = 0.25, or 25%. The complement — not drawing a heart — is 0.75, and the odds in favor are 13 : 39 = 1 : 3.
Example 2: Two Dice, Both Fours
Rolling two fair dice are independent events, so multiply: P(both 4) = (1/6) × (1/6) = 1/36 ≈ 0.0278, about a 2.78% chance.
Example 3: Overlapping Events
On a single die roll, what is P(even or greater than 4)? The events overlap at 6, so use the general addition rule: P = 3/6 + 2/6 − 1/6 = 4/6 = 2/3 ≈ 0.667. Adding 3/6 + 2/6 without subtracting the overlap double-counts the 6 — the classic error.
Example 4: Counting with Combinations
How many committees of 2 can be chosen from 5 people? Order does not matter: C(5, 2) = 5! / (2! × 3!) = 10. If the two people instead win gold and silver (order matters), it is a permutation: P(5, 2) = 5 × 4 = 20. The rule of thumb: if swapping two selected items changes the outcome, count permutations; otherwise combinations.
Example 5: Bayes' Rule in Action
A rare condition affects 1% of a population. A test catches 99% of true cases (sensitivity) and correctly clears 99% of healthy people (specificity). A patient tests positive — what is the probability they actually have the condition?
Bayes' rule combines both: P(condition | positive) = (0.99 × 0.01) / (0.99 × 0.01 + 0.01 × 0.99) = 0.0099 / 0.0198 = 0.5. Only 50% — because the condition is rare, the false positives among the healthy 99% exactly match the true positives. This is why a positive screening test is followed by more testing rather than a diagnosis.
Common Mistakes
- Counting outcomes that are not equally likely: the sum of two dice has 11 possible values, but a sum of 7 is not 1/11 — there are 36 equally likely rolls and 6 of them total 7, so the probability is 6/36 = 1/6.
- Adding probabilities of overlapping events: only mutually exclusive events add directly. Check for overlap first.
- Multiplying without checking independence: drawing two cards without replacement is dependent — after the first draw, 51 cards remain, so P(second | first) has changed.
- Confusing probability and odds: 0.25 probability is 1 : 3 odds in favor, not 1 : 4.
- Assuming "random" means "equally likely": random selection is only uniform if the mechanism guarantees it. A rigged die is random but not fair.