In a previous post (The Monty Hall Dilemma: Part 1 – Intelligence and IQ ) I explained, in some detail, the meaning and origin of the Monty Hall Dilemma that had sparked widespread public controversy in 1988, following an answer that Marilyn vos Savant posted in her Ask Marilyn column in The New York Times. So, if you are unfamiliar with that problem, just click the link above and return to this site.
In that post, I shared the correct answer and the mathematical reasoning that proves the answer, as well as some of the irate responses to Marilyn’s post, including angry responses from some Ph.D.s. Verbally animated responses to my first post prompted the creation of a second post (The Monty Hall Dilemma: Part 2 – Intelligence and IQ ) containing letters from some other perspectives, including a class of elementary students who verified the correct answer by simulation.
The discussion of the Monty Hall Problem is so productive in helping students challenge their intuition, that I subsequently included this topic in my lectures and obtained permission from Marilyn vos Savant to include her letters in my books on statistics.
The reason that the Monty Hall Dilemma is such a wonderful teaching instrument is that it shows, in dramatic fashion, that our intuition is often flawed and rational thinking is the only way to get to a correct answer. In his famous book, Thinking, Fast and Slow, Daniel Kahneman described two modes of thought that he called system 1 (fast and frugal intuitive thought) and system 2 (slow rational deductive thought). Both kinds of thinking are important to us; sometimes we need a quick, intuitive answer, and in other cases we need a carefully reasoned response.
The Monty Hall Dilemma asks us to choose which of 3 doors hides the car. So, everyone correctly assumes that the probability of choosing the door that hides the car is 1/3. After the host opens one door, revealing that the car is not behind it, our intuition tells us that merely opening that door didn’t change the probability that the car is behind either of the unopened doors. However, the fact that one door has been eliminated is new information added to the system and that changes the probabilities. When the contestant is asked whether they want to stick or switch, their are only two unopened doors, so that total probability of these two events must total 1 (i.e., it is certain that the car is behind one of the doors). The probability that the door originally chosen hides the car remains at 1/3, so the probability that the remaining door hides the car must be 2/3.
The “takeaway message” from this dilemma asserts that our intuition is not good at assessing probabilities. That’s why casinos and lotteries thrive. When the first prize in a lottery hits a large number, the multitudes flock to buy tickets, not realizing that the probability of their winning is virtually zero. The traditional response is, “Yes, but someone’s got to win.”
To test your friends on their probabilistic intuition pose the following question: What is the probability that in a randomly chosen group of 24 people, there are at least 2 people with the same birthday? (i.e., born on the same day and month, but not necessarily the same year). Many of you will recognize this as the coincident birthday problem. For those who haven’t seen this, I think you will find it engaging. An explanation and mathematical proof can be found at The Coincident Birthday Problem – Intelligence and IQ
As our brains evolved, we developed remarkable insights and deep intuitions, but our cerebral cortex had to evolve to help us navigate through probabilistic thinking.
