## INTRODUCE

###### PRESENTATION: Monty Hall

In 1990, a reader posted a question in a magazine column which confused a number of readers to the point where many did not believe the correct solution even after it had been revealed. It is a famously counterintuitive problem that has confused many people for a long time.

It is a fantastic introduction to probability as it highlights the importance of understanding probability by highlighting a key misconception as well as giving an introduction to the basic mechanics of how we calculate probability.

This presentation can be used to demonstrate the game show to your class. If you’re feeling particularly adventurous, get one of the students to play the game show by taking some boxes (and goats) into the classroom with you.

## APPLY

###### PRESENTATION: The Prosecutor’s Fallacy

Probability is a vital life skill to understanding risks. It can also have major ramifications if not understood properly in criminal law. The prosecutor’s fallacy is a widely used example of how not understanding probabilities properly can have serious repercussions.

This presentation introduces the idea of the prosecutor’s fallacy through a fictional story of a criminal case held against a man accused of stealing a woman’s bag. It highlights one example of why it is important that we have an understanding of and apply probability correctly in the real world.

You could show the presentation to the entire class or get a few of the students to act out some of the roles to add an extra layer of realism.

## AMAZE

###### PRESENTATION: The Birthday Problem

The birthday problem is a widely used problem that, similar to the Monty Hall problem, is rather counterintuitive. The question is posed: *What is the smallest number of people you need to have in a room before it is more likely than not that two people share a birthday?*

Many students will answer 182 or 183 because 182.5 is half of 365, which seems correct. No student I have ever shown this problem to has ever come close to the answer. Full disclosure – I did not get anywhere near to the answer when I first saw it!

The presentation above runs through the problem and some of the initial calculations that allow us to find the answer. Attached on one of the slides is a spreadsheet that has all of the calculations and values needed to fully understand it. With a higher level group, you can try to get them to work it out themselves (which may need scaffolding from you, perhaps talk them through the calculations on the slides and get them to work out the rest themselves). For other students, it may just be worth going through it step-by-step with them as the answer is just as amazing whether you have actually done the calculations or not.

Numberphile have a video that attacks the problem from the other direction. They ask: *What is the probability that 2 people out of a possible 23 share a birthday?* Showing this may be a good alternative to going through the presentation or can be used to compliment it.