Event Likelihood
Last reviewed: May 2026
Probability theory quantifies uncertainty. A probability of 0 means an event is impossible, 1 means certain, and values in between reflect likelihood.[1] Understanding probability helps with risk assessment, statistical analysis, games, insurance, and decision-making under uncertainty. For counting arrangements and selections, see the Permutation & Combination Calculator.
| Event | Probability | Odds |
|---|---|---|
| Coin flip (heads) | 50% | 1:1 |
| Rolling a 6 | 16.7% | 1:5 |
| Drawing an ace | 7.7% | 1:12 |
| Royal flush (poker) | 0.00015% | 1:649,739 |
| Birthday match (23 people) | 50.7% | ~1:1 |
| Winning Powerball | 0.0000003% | 1:292M |
Probability quantifies uncertainty on a scale from 0 (impossible) to 1 (certain), or equivalently from 0% to 100%. The probability of an event equals the number of favorable outcomes divided by the total number of equally likely outcomes. A fair six-sided die has a 1/6 probability (16.67%) of landing on any specific number, a 3/6 probability (50%) of landing on an even number, and a 6/6 probability (100%) of landing on some number between 1 and 6. These foundational rules scale to every probabilistic analysis — from card games to clinical trials to insurance pricing.
| Event | Probability | Odds |
|---|---|---|
| Coin flip (heads) | 50% | 1:1 |
| Rolling a 6 | 16.67% | 1:5 |
| Drawing an Ace | 7.69% | 1:12 |
| Royal flush (poker) | 0.00015% | 1:649,739 |
| Lottery jackpot (Powerball) | 0.0000003% | 1:292,201,338 |
The addition rule calculates the probability that event A or event B (or both) occurs. For mutually exclusive events (events that cannot happen simultaneously), the formula is simply P(A or B) = P(A) + P(B). Rolling a 2 or a 5 on a die: 1/6 + 1/6 = 2/6 = 33.3%. For non-mutually exclusive events (events that can overlap), you must subtract the probability of both occurring to avoid double-counting: P(A or B) = P(A) + P(B) − P(A and B). The probability of drawing a heart or a queen from a standard deck: 13/52 + 4/52 − 1/52 = 16/52 = 30.8%, subtracting the queen of hearts which was counted in both groups.
The multiplication rule calculates the probability that both event A and event B occur. For independent events (where one outcome does not affect the other): P(A and B) = P(A) × P(B). Flipping two heads in a row: 1/2 × 1/2 = 1/4 = 25%. For dependent events (where the first outcome changes the probabilities for the second): P(A and B) = P(A) × P(B|A), where P(B|A) is the conditional probability of B given that A has occurred. Drawing two aces from a deck without replacement: 4/52 × 3/51 = 12/2,652 = 0.45%. The second probability changes from 4/52 to 3/51 because one ace and one card have been removed.
Conditional probability measures the likelihood of an event given that another event has already occurred: P(A|B) = P(A and B) / P(B). Bayes' theorem extends this concept to update probabilities as new information becomes available: P(A|B) = P(B|A) × P(A) / P(B). In medical testing, Bayes' theorem explains why positive results from screening tests may still have high false-positive rates. If a disease affects 1 in 1,000 people and a test has 99% sensitivity and 95% specificity, a positive result still has only a 2% chance of indicating actual disease — because the 5% false-positive rate applied to 999 healthy people produces far more false positives than the 99% detection rate produces true positives from the 1 affected person.
Expected value (EV) is the average outcome over many repetitions of a random event, calculated by multiplying each outcome by its probability and summing the results. For a fair die: EV = (1 × 1/6) + (2 × 1/6) + (3 × 1/6) + (4 × 1/6) + (5 × 1/6) + (6 × 1/6) = 3.5. No single roll will ever yield 3.5, but over thousands of rolls, the average will converge to 3.5. Expected value drives rational decision-making in gambling, investing, insurance, and business. A lottery ticket costing $2 with a 1-in-10-million chance of winning $5 million has an EV of −$1.50 per ticket — meaning on average, you lose $1.50 every time you play.
The law of large numbers states that as the number of trials increases, the observed frequency of an event converges toward its theoretical probability. Flipping a coin 10 times might yield 7 heads (70%), but flipping it 10,000 times will almost certainly yield close to 50% heads. This principle explains why casinos are profitable — individual gamblers experience variance, but over millions of bets, the house edge plays out with mathematical precision. It also explains why insurance companies can price policies accurately despite not knowing which specific policyholders will file claims — across thousands of policies, the aggregate claim rate converges to a predictable value.
The gambler's fallacy — believing that past random outcomes influence future ones — is the most pervasive probability error. After flipping 10 heads in a row, the probability of the next flip being heads remains exactly 50%. The coin has no memory. A related error is the hot-hand fallacy in sports, though recent research suggests that some streakiness may be real in skilled performance contexts. Another common mistake is base rate neglect — ignoring the background frequency of an event when evaluating new evidence. If a rare disease test is 99% accurate but the disease affects only 1 in 100,000 people, a positive result is overwhelmingly likely to be a false positive, not a true diagnosis.
Understanding probability transforms everyday decision-making. Weather forecasts, medical test results, investment returns, and insurance policies all communicate probabilistic information. A 30% chance of rain does not mean it will rain for 30% of the day — it means that in historical situations with similar atmospheric conditions, rain occurred 30% of the time. An investment with a 12% average annual return and 20% standard deviation has a roughly 16% chance of losing money in any given year. Understanding these probabilities allows you to make better decisions about carrying an umbrella, pursuing additional medical tests, diversifying investments, and choosing appropriate insurance coverage levels.
A probability distribution describes all possible outcomes of a random variable and their associated probabilities. The normal distribution (bell curve) is the most important in statistics — it describes everything from human heights to manufacturing tolerances to stock market returns. In a normal distribution, 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. The binomial distribution applies to fixed numbers of independent yes/no trials — like the probability of getting exactly 7 heads in 10 coin flips. The Poisson distribution models rare events over time — such as the number of website visitors per minute or accidents per month at an intersection. Each distribution has specific mathematical properties that make it suitable for different types of real-world modeling.
→ P(A or B) = P(A) + P(B) - P(A and B). The addition rule for overlapping events.[1]
→ Use the complement for 'at least one.' 1 minus the probability of zero successes is much easier.
→ Independent ≠ mutually exclusive. Independent means one event does not affect the other; mutually exclusive means both cannot happen simultaneously.
→ Simulations help build intuition. Run mental or physical experiments to verify your calculations.
See also: Statistics · Permutations · Percentage · Random Number