Binomial Probability (Bernoulli Trials)

Bernoulli Trials

A Bernoulli Trial is a random experiment that repeats a single action identically over and over again. It has exactly two possible outcomes: "success" and "failure". The probability of success is the same every time the action is repeated.

Note: In Bernoulli Trials the repeated actions must be independent. Independent events are events that are not affected by other events. Let A and B represent events, A and B are independent if and only if $P(A\mid B) = P(A)$

The Binomial Probability Formula is used to find the probabilities for Bernoulli Trials:

$$P(k) = \binom{n}{k}p^kq^{n-k}$$

where, $P(k)$ = the probability of the Bernoulli Trial, $n$ = number of trials, $k$ = number of successes in $n$ trials, $n-k$ = number of failures, $p$ = probability of success in one trial and $q=1-p$ = probability of failure in one trial

Example: Flipping a Coin

We toss a coin 20 times what is the probability of coin landing on heads exactly 15 times.

We know that, $n$ = 20, $k$ = 15, $n-k$ = 5, $p$ = 0.5 = probability of the coin landing on heads and $q$ = 0.5 = probability of the coin landing on tails.

Therefore, the probability the coin lands on heads exactly 15 times is...

$$P(15) = \binom{20}{15}(0.5)^{15}(0.5)^{5} \approx 0.01478$$

Test Yourself!

Binomial Probability

There is a simple formula using a combination.

Imagine you are doing a series of experiments. Each experiment has several trials. Each trial has only two possible outcomes, usually called success or failure. This group of trials is called a Bernoulli process. Each trial is independent, which means one result does not affect the next result.

In these trials, we are interested in a certain event. For example, imagine flipping a coin and looking for heads. Each coin flip is one trial in our Bernoulli process. We can use a random variable to represent the result. We give it a value of $1$ for a success, such as heads. We give it a value of $0$ for a failure, such as tails.

Now, let's say we complete a fixed number of trials, called $n$ trials. We want to know how many times our desired event happens. This number of successes is called a binomial random variable. In other words, we are counting the successes in a fixed number of independent Bernoulli trials. This variable follows a binomial distribution.

This setup helps us study the results of these experiments. We can use it to calculate the probability of getting a certain number of successes.

(Video) Bernoulli Trials

by Wrath of Math

In this video by Wrath of Math, you will learn about Bernoulli Trials.