28 MARCH 2019

Using probability formulae for a geometric distribution

Teaching about the geometric distribution is likely to be new for your students and may also be new for you to teach. Its inclusion on the syllabus for Cambridge International AS & A Level Mathematics Probability and Statistics for examination from 2020 is, perhaps, the most significant change from the previous syllabus. 

It's important to note that there are, in fact, two discrete probability distributions that are referred to as ‘geometric’.

The first of these concerns a discrete random variable, X, which is the number of trials up to and including the first successful outcome. X can take integer values 1, 2, 3, ... with no upper limit.

The second of these concerns a discrete random variable, Y, which is the number of trials that result in failure before the first successful outcome. Y can take integer values 0, 1, 2, 3, ... with no upper limit.

In a situation where a coin is tossed repeatedly until the first head is obtained, suppose the series of results is T, T, T, T, H.

The first success occurs on the fifth trial, so X = 5.                                                                        

There are four failures before the first success, so Y = 4.

It is always the case that Y = X – 1.

https://en.wikipedia.org/wiki/Geometric_distribution has more details.

Chapter 7 of the Cambridge International AS & A Level Mathematics: Probability and Statistics 1 Coursebook contains a section on the first of these geometric distributions, i.e. X, the number of trials up to and including the first successful outcome.

In this part of the coursebook, students learn about the conditions needed to model a situation for a discrete random variable by a geometric distribution; to use three probability formulae, and about the mode and the mean/expectation of a geometric distribution.

In the Explore 7.3 feature on page 180 of the Probability and Statistics 1 Coursebook, students are challenged to derive the mean of the geometric distribution using algebraic methods.

The presentation here looks at how each of the three probability formulae can be used to solve the same problem, and briefly discusses their advantages and disadvantages. The most appropriate point at which to offer this presentation would be after the completion of Exercise 7C.

Two activities at the end of the presentation give learners the opportunity to evaluate the mode and mean in a practical context, and then to compare these with the theoretical values given by a geometric model. It would be appropriate to embark upon one or both of these activities before or after Exercise 7D. 

 

 
To view the PowerPoint, click here.