Answer: Enter your values and the Probability Calculator returns the exact result instantly — formula, worked example, and a plain-English explanation are included below the tool.
Calculate probability of single and multiple events.
Probability quantifies how likely an event is, on a scale from 0 (impossible) to 1 (certain). This calculator handles the two core combinations: the chance that two independent events both happen (multiply their probabilities) and the chance that at least one of two events happens (the union, computed as P(A) + P(B) − P(A)×P(B) for independent events).
Enter probabilities as decimals between 0 and 1 — 0.5 for a coin flip, 1/6 ≈ 0.1667 for a die face — or as percentages. The tool reports both the combined probability and its complement, the chance the event does not occur, since most real questions are answered by looking at both sides.
Two events are independent when one outcome has no effect on the other — successive coin flips, dice rolls, or draws with replacement. For independent events, P(A and B) = P(A) × P(B): two consecutive heads is 0.5 × 0.5 = 0.25. Dependence changes the rule to conditional probability, P(A and B) = P(A) × P(B|A), where the second factor is the probability of B given that A occurred.
For 'or' questions, the addition rule subtracts the overlap so it is not counted twice: P(A or B) = P(A) + P(B) − P(A and B). With mutually exclusive events — a single die cannot show both 3 and 5 — the overlap is zero and you simply add: P(3 or 5) = 1/6 + 1/6 = 1/3.
The table below collects the probabilities most often needed, from single dice to poker hands. Poker probabilities are computed for a five-card hand from a standard 52-card deck — the number of distinct hands is C(52,5) = 2,598,960, which is the denominator for every row. Note how quickly probabilities fall: three of a kind is roughly one hand in 47, while a royal flush is one in nearly 650,000.
The birthday problem is the famous counterintuitive entry: in a group of just 23 people, the chance that two share a birthday exceeds 50% — not because anyone's birthday is special, but because there are 23 × 22 ÷ 2 = 253 different pairs, each with a 364/365 chance of not matching. Small per-pair chances accumulate across many pairs.
| Event | Probability | As odds |
|---|---|---|
| Coin flip: heads | 1/2 = 50% | 1:1 |
| Die roll: a specific face | 1/6 ≈ 16.67% | 1:5 |
| Two dice: sum of 7 | 6/36 ≈ 16.67% | 1:5 |
| Two dice: sum of 12 | 1/36 ≈ 2.78% | 1:35 |
| Shared birthday in a group of 23 | 50.73% | ≈1:1 |
| Poker: three of a kind | ≈2.11% | ≈1:46 |
| Poker: flush | ≈0.197% | ≈1:508 |
| Poker: royal flush | 1/649,740 ≈ 0.000154% | 1:649,739 |
Probability and odds are different currencies. A probability of 0.25 equals odds of 1:3 (one way to win, three to lose) and decimal odds of 4.0 — a fair payout of $4 per $1 staked. Converting matters wherever gambling, insurance, or betting markets quote odds: decimal odds of 4.0 imply a 25% chance, and markets with a margin quote implied probabilities summing above 100%.
Expected value multiplies each outcome by its probability and sums: a 25% chance of winning $300 and 75% chance of losing $50 has EV = 0.25 × 300 + 0.75 × (−50) = $37.50 per play. Positive EV does not guarantee a win on any single trial — it describes the long-run average, which is why a positive-EV bet can still lose money for a long stretch before the average asserts itself.
How do I calculate the probability of two events both happening?
If the events are independent, multiply: P(A and B) = P(A) × P(B). Two consecutive heads is 0.5 × 0.5 = 0.25. If the events are dependent, replace the second factor with the conditional probability P(B given A) — the chance of B after accounting for A having happened.
How do I calculate the probability of either of two events?
Add the probabilities, then subtract the chance of both occurring so the overlap is not double-counted: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events like distinct die faces, the overlap is zero and you simply add.
What is the difference between probability and odds?
Probability is the chance of success out of all trials; odds compare success to failure. A probability of 0.25 is odds of 1:3, or 1-to-3 against. Decimal odds of 4.0 correspond to 25% probability. The calculator works in probabilities; divide p by (1−p) to express a result as odds.
Can a probability be greater than 1?
No. Probability runs from 0 (impossible) to 1 (certain) by definition, which is also why probabilities of all possible outcomes must sum to exactly 1. If your computed combination exceeds 1, a premise is wrong — most often treating dependent events as independent, or adding non-exclusive events without subtracting their overlap.
What is the birthday paradox?
The surprising fact that a shared birthday becomes more likely than not in a group of just 23 people (probability ≈ 50.7%, ignoring February 29). It feels wrong because we think about one person's birthday rather than the 253 distinct pairs a group of 23 contains, each pair a separate chance to match.
How is expected value different from probability?
Expected value weights each outcome by money or utility, not just likelihood: multiply each outcome's value by its probability and sum. A 25% chance to win $300 and 75% chance to lose $50 has EV of +$37.50. It predicts the long-run average, not any single result — which is why short-term outcomes routinely defy a positive EV.