Bernoulli trial

Bernoulli Trial

A Bernoulli trial is a random experiment in which there are only two possible outcomes: success or failure. The probability of success is denoted by p and the probability of failure is denoted by q, where q = 1 - p. Bernoulli trials are named after the Swiss mathematician Daniel Bernoulli who made significant contributions to the field of probability and statistics in the 18th century.

Definition

A Bernoulli trial must satisfy the following conditions:

• The experiment must have only two possible outcomes, often termed as "success" and "failure".
• Each trial is independent of the others, meaning the outcome of one trial does not affect the outcome of another.
• The probability of success p remains constant for each trial.

Mathematical Representation

The outcome of a Bernoulli trial can be mathematically represented by a Bernoulli distribution, which is a discrete probability distribution. If X is a random variable that represents the outcome of a Bernoulli trial, then X can take the value of 1 with probability p (indicating success) and the value of 0 with probability q = 1 - p (indicating failure). The probability mass function (PMF) of X is given by:

\[P(X=x) = p^x(1-p)^{1-x}\]

for x = 0, 1.

Applications

Bernoulli trials are foundational in the field of probability and statistics and have numerous applications. They are used in the modeling of binary outcomes in various disciplines including medicine, psychology, engineering, and finance. For example, in clinical trials, the effectiveness of a new medication can be considered a Bernoulli trial where success is defined as the medication having the desired effect and failure is defined as it not having the desired effect.

Related Concepts

• Binomial Distribution: If a series of n independent Bernoulli trials are performed, and the number of successes is counted, the distribution of the number of successes is a binomial distribution.
• Law of Large Numbers: This law states that as the number of Bernoulli trials increases, the sample proportion of successes will tend to approach the theoretical probability of success p.
• Bernoulli Process: A sequence of Bernoulli trials is known as a Bernoulli process.

See Also

• Probability Theory
• Random Variable
• Discrete Distribution

External Links

• Wikimedia Commons Category: Bernoulli Trials

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