Sense of Probability:
Fallacies, Myths and Puzzles
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
Also, we now have a regularly updated blog addressing similar themes: Probability
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
affect your life. And
would expect that, where critical decisions need to be made,
probabilities are calculated correctly. Unfortunately, they
often not. In medical and legal sitatuations lives are affected as a
result. In business, companies can go bust and in many
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
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
have provided input, comments and corrections. They include: Anthony
Constantinou, Steve Horn, Martin Kryzwinski, Keith Lee, Peter
McOwan, Kristoffer Spicer, Jeremy Stone.