Table of Contents

## Introduction to Bernoulli Trials

A Bernoulli trial is a process that can result in one of two outcomes, typically denoted as success or failure. The two outcomes are independent of each other, meaning that the probability of one outcome does not influence the probability of the other.

The Bernoulli trial is a fundamental concept in probability theory, and is used in a variety of settings, including gambling and statistical sampling.

## Bernoulli Trials

A Bernoulli trial is a random event with two outcomes, “success” and “failure”, that are mutually exclusive and exhaustive. The probability of success is constant for all trials.

## Definition

of a Simile

A simile is a figure of speech that compares two different things, usually by using the word “like” or “as.” For example, “My love for you is like a rose; strong and beautiful, but with thorns that can hurt.”

## Binomial Distribution

The binomial distribution is a discrete probability distribution that gives the probability of getting a certain number of successes in a sequence of n independent Bernoulli trials. The binomial distribution is often used to model the number of successes in a sequence of Bernoulli trials, such as the number of heads in a sequence of coin flips.

The binomial distribution has the following properties:

1. The binomial distribution is a discrete distribution.

2. The binomial distribution has two parameters: n and p.

3. The binomial distribution is a bell-shaped curve.

4. The binomial distribution is symmetric about its mean.

5. The binomial distribution has a peak at its mean.

6. The binomial distribution is approximately Normal when n is large and p is small.

7. The binomial distribution is a truncated distribution.

8. The binomial distribution is a bounded distribution.

9. The binomial distribution has a PDF of:

where:

n is the number of successes

p is the probability of success

x is the number of successes

## More About Binomial Distribution

A binomial distribution is a type of probability distribution that describes the outcome of a series of n independent and identically distributed Bernoulli trials. In each trial, the probability of success is p, and the trials are independent of each other.

## Bernoulli Trial

A Bernoulli trial is a random event that has two possible outcomes, success or failure. The probability of success is the same for each trial.

## Bernoulli Trials Example

A fair coin is flipped 10 times. What is the probability of getting exactly 5 heads?

The probability of getting 5 heads is 1/2.

## Bernoulli Trials Conditions

The Bernoulli trials conditions are a set of mathematical conditions that must be met in order for a sequence of events to be a series of Bernoulli trials. The conditions are:

The events must be independent.

The events must have two possible outcomes, which are called “success” and “failure”.

The probability of success must be the same for each event.

The number of trials must be finite.

## Bernoulli Trials Formula

The Bernoulli Trials Formula is a mathematical formula used to calculate the probability of a certain outcome in a series of Bernoulli trials. The formula is:

P(x) = (1/2)^x

Where “x” is the number of times the event has occurred.

## Bernoulli Trials and Binomial Distribution

A Bernoulli trial is a random experiment with two possible outcomes, usually denoted as success and failure. The binomial distribution is a probability distribution that describes the results of a Bernoulli trial.

The binomial distribution has the following properties:

The probability of success is p.

The probability of failure is q = 1 – p.

The number of successes (n) is a discrete variable that can take on the values 0, 1, 2, …

The probability of getting exactly k successes is given by the binomial distribution function:

B(k; n, p) =

where

The binomial distribution is symmetric about the mean, which is

E(x) = np.

The variance is given by

Var(x) = np(1 – p).

## Bernoulli Distribution

A Bernoulli distribution is a type of discrete probability distribution that can be used to model the outcomes of a coin flip or the success or failure of a process.

## Bernoulli Distribution Examples

Example 1

A coin is tossed 10 times. What is the probability of getting 3 heads?

The probability of getting 3 heads is 0.3.

## Properties of Bernoulli Distribution

The Bernoulli distribution is a discrete probability distribution.

The Bernoulli distribution has a single parameter, p, which is the probability of success.

The Bernoulli distribution is a symmetric distribution.

The Bernoulli distribution is a memoryless distribution.