Probability Distribution Calculator

A binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It is defined by two parameters: n (number of trials) and p (probability of success on each trial). Classic examples include flipping a coin n times or testing n products for defects.

A binomial experiment has four key properties: it consists of a fixed number of trials (n), each trial has exactly two possible outcomes (success or failure), the probability of success (p) is constant across all trials, and the trials are independent of each other. Rolling a die 20 times and counting sixes is a classic binomial experiment.

Binomial probability is computed using the formula P(X = x) = C(n, x) × p^x × (1−p)^(n−x), where C(n, x) is the binomial coefficient (n choose x), p is the probability of success, and x is the number of successes. Cumulative probabilities like P(X ≤ x) are found by summing P(X = k) for all k from 0 to x.

P(X = x) gives the exact probability that the random variable X equals a specific value x. P(X ≤ x) is the cumulative probability that X takes any value from 0 up to and including x. Cumulative probabilities are useful when you want to know the chance of obtaining at most a certain number of successes.

A normal distribution is a continuous bell-shaped probability distribution defined by its mean (μ) and standard deviation (σ). It is symmetric around the mean, with about 68% of values falling within one standard deviation, 95% within two, and 99.7% within three. Many natural phenomena, such as heights and test scores, follow a normal distribution.

Standard deviation (σ) measures how spread out the values in a distribution are around the mean. A small standard deviation means values cluster tightly around the mean, while a large standard deviation indicates more variability. For a binomial distribution, σ = √(n × p × (1−p)).

For a normal distribution, P(X ≤ x) is computed using the cumulative distribution function (CDF), which integrates the normal probability density function from −∞ to x. This is expressed as Φ((x − μ)/σ), where Φ is the standard normal CDF. This calculator uses an accurate numerical approximation of the error function to compute these values.

The probability of success on a single trial (p) is the likelihood that one specific trial results in the desired outcome. It must be a value between 0 and 1, where 0 means the event never occurs and 1 means it always occurs. For example, if you flip a fair coin, the probability of heads on a single flip is 0.5.