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Probability Calculator

Single events · AND/OR/Conditional · Binomial · Normal distribution · Dice/Card/Coin

Quick Examples

Probability Gauge 0%
0% — Impossible 50% 100% — Certain

Formula: P = Favorable Outcomes ÷ Total Outcomes

Quick Formulas Reference

P(A) = Favorable / Total
P(A') = 1 − P(A)
P(A∩B) = P(A) × P(B) [independent]
P(A∪B) = P(A) + P(B) − P(A∩B)
P(A|B) = P(A∩B) / P(B)
Binomial = C(n,k) × pk × (1−p)n−k
Z-score = (X − μ) / σ
C(n,k) = n! / (k! × (n−k)!)

Worked Examples

Lottery Odds

A lottery requires picking 6 numbers from 1–49. Favorable = 1, total = C(49,6) = 13,983,816. Probability = 1/13,983,816 ≈ 0.0000000715 (7.15 × 10⁻⁸). The complement (not winning) = 99.99999285%.

Quality Control (Binomial)

A factory has a 5% defect rate (p = 0.05). In a batch of n = 20 items, the probability of exactly k = 2 defects is C(20,2) × 0.05² × 0.95¹⁸ ≈ 0.1887 (18.87%). P(at most 2) ≈ 92.45%.

Exam Scores (Normal)

Exam scores are normally distributed with μ = 70, σ = 10. P(score > 85) = P(Z > 1.5) ≈ 6.68%. P(60 < X < 80) = P(−1 < Z < 1) ≈ 68.27% (empirical rule).

What Is Probability?

Probability is a branch of mathematics that quantifies uncertainty. It measures how likely an event is to occur on a scale from 0 (absolutely impossible) to 1 (absolutely certain). The basic probability formula divides the number of favorable outcomes by the total number of equally likely outcomes: P = Favorable / Total. A fair coin has a 1/2 = 0.5 = 50% chance of landing heads.

Types of Probability Events

Events can be classified in several ways that affect how probabilities are combined:

  • Independent events: The occurrence of one does not change the probability of the other. Example: two separate coin flips.
  • Mutually exclusive events: Both cannot occur simultaneously. Example: rolling a 3 and a 5 on the same die roll.
  • Complementary events: Together they cover all possibilities. P(A) + P(A') = 1.
  • Conditional events: The probability of A given that B has already occurred, written P(A|B).

The Addition Rule (OR)

The probability that event A or event B occurs is: P(A∪B) = P(A) + P(B) − P(A∩B). The intersection P(A∩B) is subtracted because it would be counted twice. For mutually exclusive events where P(A∩B) = 0, this simplifies to P(A∪B) = P(A) + P(B).

The Multiplication Rule (AND)

For independent events, the probability that both A and B occur is: P(A∩B) = P(A) × P(B). For dependent events, you need the conditional form: P(A∩B) = P(A|B) × P(B). For three independent events: P(A∩B∩C) = P(A) × P(B) × P(C).

Binomial Distribution

The binomial distribution applies when you have a fixed number of independent trials (n), each with only two outcomes (success or failure), and the same probability of success (p) on every trial. The probability of exactly k successes is:

P(X = k) = C(n,k) × pk × (1−p)n−k

MetricFormula
Meanμ = np
Varianceσ² = np(1−p)
Standard Deviationσ = √(np(1−p))
P(X ≤ k) CDFSum of P(X = i) for i from 0 to k
P(X ≥ k)1 − P(X ≤ k−1)

Normal Distribution and Z-Scores

The normal distribution is the famous bell-shaped curve described by its mean (μ) and standard deviation (σ). Any value X can be converted to a Z-score to express how many standard deviations it is from the mean: Z = (X − μ) / σ. Once converted, the standard normal table (or this calculator) gives the exact probability.

The Empirical Rule (68-95-99.7)

For any normal distribution:

  • About 68.27% of values fall within 1 standard deviation of the mean (μ ± σ)
  • About 95.45% fall within 2 standard deviations (μ ± 2σ)
  • About 99.73% fall within 3 standard deviations (μ ± 3σ)

This rule is invaluable for quick mental estimates. If exam scores are normally distributed with mean 70 and standard deviation 10, then about 68% of students scored between 60 and 80.

Conditional Probability and Bayes

Conditional probability P(A|B) is the probability that event A occurs given that event B has already occurred. It is defined as P(A|B) = P(A∩B) / P(B). This concept underpins Bayes' theorem, which reverses the conditioning: P(B|A) = P(A|B) × P(B) / P(A). Conditional probability is used extensively in medical testing, machine learning, and risk analysis.

Dice, Cards, and Coin Probabilities

Classic probability problems involve physical randomizers:

  • Fair die (d6): P(specific face) = 1/6 ≈ 16.67%
  • Two dice sum: P(sum=7) = 6/36 = 1/6 ≈ 16.67% (most common two-dice sum)
  • Standard deck: P(Ace) = 4/52 = 1/13 ≈ 7.69%; P(Heart) = 13/52 = 1/4 = 25%
  • Fair coin: P(exactly 5 heads in 10 flips) = C(10,5) × 0.5¹⁰ ≈ 24.61%

Frequently Asked Questions

What is probability?
Probability is a measure of how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of rolling a 6 on a fair die is 1/6 ≈ 0.1667 or 16.67%.
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other's probability. For independent events, P(A AND B) = P(A) × P(B). Mutually exclusive events cannot both occur at the same time, so P(A AND B) = 0 and P(A OR B) = P(A) + P(B). Flipping heads and tails on the same coin flip are mutually exclusive; flipping two separate coins are independent events.
How does the binomial distribution work?
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. The probability of exactly k successes in n trials is P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where C(n,k) is the binomial coefficient. The mean is np and the variance is np(1-p).
What is a Z-score in normal distribution?
A Z-score measures how many standard deviations a value X is from the mean μ. It is calculated as Z = (X - μ) / σ. A positive Z-score means X is above the mean; a negative Z-score means X is below the mean. Z-scores allow you to look up probabilities in the standard normal table or use a normal distribution calculator.
What is conditional probability?
Conditional probability P(A|B) is the probability that event A occurs given that event B has already occurred. It is calculated as P(A|B) = P(A ∩ B) / P(B). For example, if you draw a card and it is red, the conditional probability it is also a heart is P(heart|red) = (13/52) / (26/52) = 0.5.
What is the empirical rule (68-95-99.7 rule)?
The empirical rule states that for a normal distribution: approximately 68.27% of data falls within 1 standard deviation of the mean (μ ± σ), about 95.45% falls within 2 standard deviations (μ ± 2σ), and approximately 99.73% falls within 3 standard deviations (μ ± 3σ). This is invaluable for quick mental estimates of probability ranges.
How do you calculate dice probability?
For a single fair die with d sides, the probability of rolling any specific number is 1/d. For multiple dice, the probability of a specific sum is the count of combinations producing that sum divided by d^n total outcomes. This calculator uses dynamic programming to compute exact probabilities for any number of dice and sides quickly.
What is the complement rule in probability?
The complement rule states P(A') = 1 − P(A). The probability of an event NOT happening plus the probability of it happening always equals 1. This is often the easiest way to calculate "at least one" probabilities: P(at least one) = 1 − P(none at all). For example, P(at least one head in 5 flips) = 1 − (0.5)^5 = 1 − 0.03125 = 96.875%.

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