Copyright: 2008
Publisher: Pantheon Books
ISBN: 978-0-375-42404-5

Leonard Mlodinow takes us on a delightful journey through the fascinating history of probability and statistics.  On the way he manages to sneak in a very practical explanation of the basics of the field.  Mingling stories of Pascal, Fermat, Bayes and others with the Law of Large Numbers, Bayesian Probability and confidence levels, Mlodinow makes The Drunkard's Walk a coherant and entertaining read.  He tops the book off with a very practical application of what chance and probability can mean to ordinary peope like you and I.  In the process he points out that the main reason you and I are ordinary is chance!

Basic Laws of Probability

Early in the book, Mlodinow points out a number of fundamental laws of probability that can be helpful for gut checks in every day life:

  1. Probability that 2 events will both occur cannot be greater than the probability that each will occur individually.
  2. To calculate the probability that 2 independent events will both occur, you multiply their probabilities together.
  3. To calculate the probability that either of 2 independent events will occur you add their probabilities together.

Note that these laws apply to independent events, not dependent ones.  Daniel Kahneman and Amos Tversky conducted an experiment in which they asked some graduate students to consider the following statement and assign estimated probabilities:

  1. Linda is active in the feminist movement
  2. Linda is a bank teller and is active in the feminist movement
  3. Linda is a bank teller

87% of the participants ranked number 2 as a higher probability than number 3.  Without any other details about Linda's background, we know by applying Rule #1 that this cannot be the case.  Yet in our every day life we often make the same logical mistakes because our minds tend to assign dependency even where there shouldn't be any assumption made.

Let's Make a Deal... and then Switch!

If you have read much in the field of probability or game theory, you have probably run across the Monty Hall problem.  In a nutshell Let's Make a Deal consists of having three doors, behind one of them is a real prize, behind the other two there are goofy prizes.  The game consists of the player choosing a door randomly and then the host eliminates one of the two doors that is left.  Now the player must decide: without know what is behind the door he / she has chosen, should they switch to the new door or keep the one they have?

This question was put Marilyn vos Savant in her Parade magazine column "Ask Marilyn".  She explained to the reader that the player should ALWAYS switch to the remaining door.  If you are like me, the first time you heard this you thought to yourself "um... no... because it doesn't matter!"  After all, you don't know if you have the winning one or the losing one so switching really won't make a difference!

But it does... and Mlodinow does a fantastic job of explaining this conundrum in his book.  He explains it simply by pointing out that in the initial state of the game, you have  1 / 3 chance of guessing the right door.  This means that you have a 2 / 3 chance of having guessed the wrong door.  Once the host has eliminated a door, you know that particular door wasn't the one with the prize.  This means there is a 2 / 3 chance that the door you chose is the wrong one.  Since the door you have selected is most likely the wrong one, you should always switch.

Still being skeptical, my first thought was "I am going to write a simulator to prove this!"  Turns out quite a few people have already done so... some of them being available online.  One of them is by the Statistics Department of the University of South Carolina and it proves the theory right if you play with it for just a while.

Let Me Count the Ways

A very practical note to make to yourself is that the chances of an event happening vary depending on the number of ways that it can happen.  So as Galileo noted with rolling three dice, there are 27 ways of rolling a 10 but only 25 ways of rolling a total of 9.  This means that the chances of rolling a 10 is about 1.08 times more likely than rolling a 9. 

Bayesian Probability

Earlier we noted a number of laws of probability that only applied only to situations where the events were completely independent.  At times though, events are conditional at which point we enter the realm of Bayesian probability.  It is an important and practical concept to understand.

Mlodinow uses an easy to understand example.  What if the boss is taking a long time to respond to your emails?  Perhaps you are no longer in favor and need to find a new job.  IF your star is beginning to fade, there is a high probability that you need to start looking for a job.  And IF your boss is unhappy with you, he will likely be slow responding to your emails.  So we logically conclude a high probability that it is time to polish up the resume.

But wait... the reality is that there are a large number of reasons for your boss being slow to answer emails.  Perhaps he / she is busy with a large project.  Perhaps he / she has so much faith in you they don't believe you need constant baby-sitting.  Perhaps the mail server is running slow during the week.  Given that there are a large number of ways to explain this behavior, there is a lower probability that you have fallen out of favor. 

Usually Bayesian probability problems are couched in terms of medical tests that come out with false positives.  Mlodinow talks about those but I enjoyed his more applicable scenarios as they are more relevant to daily life.

Notes and Quotes
  • People have a very poor perception of randomness
  • "There is a difference between a process being random and the output of that process appearing random" - George Spencer-Brown *
  • We should judge people by their ability, not by their success.
  • "If you want to succeed, double your failure rate." - Thomas Watson

* Steve Jobs noted in a New York Times article that the iPod shuffling had to be "de-randomized" a bit because repeats caused people to think it wasn't random.  It really was more random before but they had to fake the randomness in order to make it appear more random.

Conclusion

This book has been on my reading list for a while and after reading it, I wish I had read it long ago!  Such a fascinating and delightful read.  Considering that Mlodinow is a physicist and considering that most physicists are very smart but not particularly engaging, this book was a wonderful surprise.  This book is a good read for anyone in any field who wants some practical insight to how the world works.