Bernoulli Trial Calculator

A Bernoulli trial, named after the Swiss mathematician Jacob Bernoulli, is a random experiment or statistical experiment with exactly two possible outcomes: "success" and "failure". These trials are fundamental in probability theory and statistics, often used to model simple, binary events where the result is either a positive or negative outcome.

The Bernoulli trial is a fundamental concept in probability theory, named after the Swiss mathematician Jacob Bernoulli. It represents a random experiment with two possible outcomes, often referred to as "success" and "failure."

The Basics

In a Bernoulli trial, there are only two possible outcomes, and each trial is independent of the others. Let's denote:

n: The number of trials.
k: The number of successes.
p: The probability of success in a single trial (ranging from 0 to 1).

The Formula

The probability mass function (PMF) for a Bernoulli distribution is given by the formula:

$P(X=k) = C(n,k) \cdot p^k \cdot (1-p)^{n-k}$

Where:

P(X=k): Probability of having k successes.
C(n, k): Combinations, denoting the number of ways to choose k successes from n trials.

The Bernoulli Theorem

The Bernoulli theorem describes the behavior of the sum of independent and identically distributed Bernoulli random variables. If X₁, X₂, ..., X_n are independent Bernoulli random variables with the same probability of success (p), then the sum Y = X₁ + X₂ + ... + X_n follows a binomial distribution.

Applications

Bernoulli trials find applications in various fields, including statistics, biology, and quality control. They are the foundation of the binomial distribution, which is widely used in modeling random phenomena.

Understanding the Bernoulli trial is crucial for grasping more complex probability concepts. Whether you're exploring the basics of statistics or diving into the intricacies of probability theory, the Bernoulli trial serves as a cornerstone in your mathematical journey.

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