Probability Distribution Calculator
Calculate probabilities for various discrete probability distributions.
Binomial distribution: probability of x successes in n trials with probability p of success on each trial.
P(X = x)
Understanding Probability Distributions
A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes for an experiment. It describes how the probabilities are distributed over the values of the random variable.
There are two main types of probability distributions:
- Discrete probability distributions describe the probability of occurrence of each value for a discrete random variable (a variable that can only take specific values, often integers).
- Continuous probability distributions describe the probability of occurrence of values in a continuous range for a continuous random variable.
This calculator focuses on discrete probability distributions, which are commonly used in statistics and probability theory.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.
Key Properties
- Fixed number of trials (n)
- Each trial is independent
- Each trial has the same probability of success (p)
- Each trial has only two possible outcomes (success or failure)
Formulas
PMF: P(X = x) = C(n, x) · p^x · (1-p)^(n-x)
Mean: E(X) = n · p
Variance: Var(X) = n · p · (1-p)
Examples
- Number of heads in a fixed number of coin flips
- Number of defective items in a batch of n items
- Number of successful free throws by a basketball player
Poisson Distribution
The Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming events occur with a known average rate and independently of each other.
Key Properties
- Events occur independently
- Events occur with a constant mean rate (λ)
- Events occur one at a time (no simultaneous events)
- The probability of an event in a small interval is proportional to the size of the interval
Formulas
PMF: P(X = x) = (e^-λ · λ^x) / x!
Mean: E(X) = λ
Variance: Var(X) = λ
Examples
- Number of calls received by a call center in an hour
- Number of typos on a page of a book
- Number of cars arriving at a toll booth in a minute
Geometric Distribution
The geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials.
Key Properties
- Each trial is independent
- Each trial has the same probability of success (p)
- Counts the number of trials until the first success
Formulas
PMF: P(X = x) = (1-p)^(x-1) · p
CDF: P(X ≤ x) = 1 - (1-p)^x
Mean: E(X) = 1/p
Variance: Var(X) = (1-p)/p²
Examples
- Number of coin flips needed to get a head
- Number of attempts until first success in a game
- Number of interviews conducted until finding a suitable candidate
Hypergeometric Distribution
The hypergeometric distribution models the number of successes in a sample drawn without replacement from a finite population.
Key Properties
- Sampling is done without replacement
- Population size is finite
- Population has a known number of success states
Formulas
PMF: P(X = x) = [C(K, x) · C(N-K, n-x)] / C(N, n)
Mean: E(X) = n · (K/N)
Variance: Var(X) = n · (K/N) · ((N-K)/N) · ((N-n)/(N-1))
Examples
- Number of defective items in a sample taken from a batch
- Number of white balls drawn from an urn containing white and black balls
- Number of correct answers when randomly guessing on a multiple-choice test
Glossary
PMF (Probability Mass Function)
Gives the probability that a discrete random variable is exactly equal to a value.
CDF (Cumulative Distribution Function)
Gives the probability that a random variable is less than or equal to a value.
Expected Value (Mean)
The weighted average of all possible values in the distribution.
Variance
Measures how far the values are spread out from the mean.
Combination
C(n,k) = n! / (k! * (n-k)!) = Number of ways to choose k items from n distinct items, where order doesn't matter.