Fallacies, Myths and Puzzles

Please note: Much of the material here has now been incorporated (and improved) into the book "Risk Assessment and Decision Analysis with Bayesian Networks" by Norman Fenton and Martin Neil. The book includes many more worked examples like these. The website for the book is here. Also, we now have a regularly updated blog addressing similar themes: Probability and Risk.

You have probably heard the following 'puzzle':

In a class of 23 children what is the probability that at least two children share the same birthday?

You might even remember that the answer is 'better than 1 in 2". But you probably cannot explain why the answer is anything like this, and when asked the question the first time you probably gave a completely wrong answer - one that massively underestimates the true probability.

Don't worry because most people (even highly intelligent people like world-leading barristers, scientists, surgeons and businessmen) make exactly the same mistakes (just as they can be easily convinced of many things that are totally false like this). Not knowing the probability that children share the same birthday is hardly going to affect your life. But the problem is strikingly similar to many that can and do affect your life. And you would expect that, where critical decisions need to be made, the probabilities are calculated correctly. Unfortunately, they are often not. In medical and legal sitatuations lives are affected as a result. In business, companies can go bust and in many everyday financial cases the public's general inability to understand probabilities is cunningly used against them.

And it is not just about relying on other people's ability to calculate probabilities properly. Every day you make decisions that, consciously or not, depend on probability assessments. Whether it is deciding which way to travel to work, deciding if it is worth taking out a particular insurance, deciding if you should proceed with a major project, or improving your chances of winning at cards or on a sporting bet, the ability to do accurate probability calculations is the ONLY way to ensure that you make the optimal decisions.

The examples on these page are everyday, popular examples of probability problems that people normally get wrong or do not understand. Some examples are quite well known (including well known fallacies), others are problems that people have asked me to explain or calculate. Have fun looking at them - you never know how important it might be to understand some of them

- The birthday sharing problem
- Did the prosecutor get it right?
- Did the defence get it right?
- The fallacy of reasoning about evidence in court (this is also an introduction to Bayes Theorem)
- Why
clever people cannot understand Bayes Theorem and what you can do about
it (show an event tree)

- Frequentist versus Subjective view of Uncertainty
- Why
the basics of probability are not so hard

- Stick or switch? (The Monty Hall game show problem)
- Boy
or Girl? (If you know somebody has two children, one of whom is a boy,
what is the probability the other child is a boy?)

- Did the doctor make the correct medical diagnosis?
- How can your scores be worse than mine every year but still come out higher on average than mine?
- Optical illusions
- How
certain are you of what you see? (confirmation bias)

- Which
sequence occurs first?

- Was the cup draw fixed?
- Why do 'impossible' events happen all the time?
- How comes 'likely' events happen so rarely?
- Why
is every shuffled pack of cards a miracle?

- What is the probability of seeing the same truck in two completely different cities?
- What's the probability you have cancer if you are a smoker? (this is a basic explanation of Bayes Theorem)
- Are the bookies being fair on 'double' bets?
- If half of the people who start a course pass it why can't I conclude that 100 people started the course if I know that 50 passed?
- The Schools League Table Fallacy
- Can
you improve your chances of winning the lottery with just one ticket?

- Why are you more likely to die in a car crash when the road conditions are good compared with when they are bad?
- The
confidence interval fallacy

- A
footballer or a nurse?

- Humorous stuff
- Maximum
likelihood demystified

Acknowledgements: Many people have provided input, comments and corrections. They include: Anthony Constantinou, Steve Horn, Martin Kryzwinski, Keith Lee, Peter McOwan, Kristoffer Spicer, Jeremy Stone.

Norman Fenton