What is a Binomial Distribution?

In the first tutorial I show you what a Binomial Distribution is by considering various different tree diagrams to determine the conditions. You are also introduced to the notation used to describe a random variable that is Binomially distributed.

Tutorial : What is a Binomial Distribution?

The Binomial Formula

In the next tutorial I show you how to calculate binomial probabilities without the need for drawing a tree diagram.

Tutorial : The Binomial Formula

Examples

Now I look at using the formula in solving the following questions. You may like to try these before looking at the tutorial and worked solution.

(1)

The manufacturer of a bag of sweets claims that there is a 90% chance that the bag contains some toffees. If 20 bags are chosen. What is the probability that

(i) all the bags contain toffees

(ii) more than 18 bags contain toffees

(2)

In clinical trials a certain drug has an 8% success rate of curing a known disease. If 15 people are known to have the disease. What is the probability of at least 2 being cured.

Tutorial and worked solutions to 1 and 2

Cumulative Binomial Probability Tables

There are tables in most formulae books or in the back of a statistics text book that allow you to calculate some Binomial probabilities with ease. This is outlined in the following tutorial.

Tutorial on using the Cumulative Binomial Probability Tables

Sometimes the tables only give values of p up to 0.5.

This tutorial explains how to use the tables still when p>0.5.

Tutorial on using the Cumulative Binomial Probability Tables for when p>0.5

The Mean and Variance of a Binomial Distribution

In this next tutorial you are shown what the mean and variance of a Binomial Distribution is and then asked the following questions.

(3) A normal 6 sided fair die is thrown 24 times and the number of 6's scored is recorded. Find the mean and variance.

(4) If X is a r.v. such that X~B(n,p) and E(X)=12 and Var(x)=3. Find n and p.

You may like to try these before looking at the tutorial.

Tutorial and worked solutions to 3 and 4

Poisson Approximation to the Binomial Distribuition

The Poisson distribution can be used as an approximation to the Binomial distribution providing certain conditions are met. This can greatly simplify calculations.

In the next tutorial you are shown these conditions.

Tutorial on the Poisson approximation to the Binomial distribution

Now we have an example where the approximation can be used. However, the video will compare the real answer with the approximation. You may like to try it before looking at the video and comparing your working.

A factory produces a particular electrical component and on average 1 in 50 is faulty. In a batch of 300 components taken at random, what is the probability of having at least eight faulty components?

Worked solution comparing the exact and approximate probabilities.