I would like to invite you to take part in a 1980’s TV game show called Let’s Make A Deal. The host of this show, Monty Hall shows you three closed doors. Behind one of the doors is a luxury car. Behind the other two doors are goats. Your objective is to try and win the car by picking the door that the car is behind. The host of the show knows which door that is.
Let us say you pick door B, the middle one. Monty Hall then opens door C to reveal a goat. The car must be either behind door A or the one that you picked i.e. door B.
Now the host gives you a choice. You can stay with your initial choice of door B or switch to door A. What will you do?
It is important to realise that the host knows which door the car is behind so no matter which door you choose he will always chose another door with a goat behind it. The object of the game is to get you to change your mind about your first choice.
If you are like most people you probably won’t want to switch. A cognitive bias known as endowment effect will make you want to stick to your first choice. Even when the host offers you inducements to switch you will still probably resist. After all, is there not a 50-50 chance that the car is behind either door? The host may just be trying to trick you out of your valuable prize.
In 1990, a reader of Parade magazine wrote to the author of their “Ask Marilyn” column asking , whether it was better to stay with the original choice or switch. Marilyn’s advice was to switch. This caused an uproar. Parade magazine’s columnist, Marilyn vos Savant is recognised as a lady with a very high IQ but more than 10,000 readers, including about 1000 with PhD’s wrote to the magazine angrily claiming that she was wrong.
Here are a few of the letters –
You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country. Shame!
Scott Smith, Ph.D.
University of Florida
May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?
Charles Reid, Ph.D.
University of Florida
I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns.
W. Robert Smith, Ph.D.
Georgia State University
You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively towards the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind?
E. Ray Bobo, Ph.D.
Georgetown University
You made a mistake, but look at the positive side. If all those Ph.D.’s were wrong, the country would be in some very serious trouble.
Everett Harman, Ph.D.
U.S. Army Research Institute
You are the goat!
Glenn Calkins
Western State College
Maybe women look at math problems differently than men.
Don Edwards
Sunriver, Oregon
But Marilyn was right and you don’t need a PhD in mathematics to prove it, in fact you don’t really need much in the way of maths at all. You do however need to use your system 2 thinking that I discussed in previous articles – Market Blindness part 1 & 2. Here is the solution.
There are only 9 possible situations which are shown in the first 2 columns. The 3^{rd} column shows the host’s possible actions and the last 2 columns show the result of switching or not switching. By switching it can be clearly seen that you have twice as much chance of winning.
Car behind door |
Contestant chooses door |
Monty Hall opens door |
Result (no switch) |
Result (switch doors) |
A |
A |
B or C |
Win |
Lose |
A |
B |
C |
Lose |
Win |
A |
C |
B |
Lose |
Win |
B |
A |
C |
Lose |
Win |
B |
B |
A or C |
Win |
Lose |
B |
C |
A |
Lose |
Win |
C |
A |
B |
Lose |
Win |
C |
B |
A |
Lose |
Win |
C |
C |
A or B |
Win |
Lose |
Why did so many highly qualified people get the answer wrong and get so upset about Marilyn’s correct answer? I would suggest the reason is because they were blind-sided by their system 1 thinking which can often cause us to jump to conclusions without looking at the real problem. They were ignoring the readily available information that the shows’ host, Monty Hall, was changing probability by the way he was running the game. Instead they relied totally on their classical text book learning which didn’t take a situation like this into account. As for getting upset, this is a normal response of system 1 thinking. It tends to defend its unconscious decisions using what is commonly known as rationalisation. You don’t need a PhD to think and act in this way. We all do it. It is part of our evolutionary mental heritage.
Problems like the Monty Hall problem occur every day in investment and trading. We need to develop and use our system 2 thinking in conjunction with the executive function in our brain to deal with these problems effectively. Otherwise, just like all those PhD experts we will fall back to habitual thinking and miss vital clues that can significantly impact our trading and profits. Furthermore, financial markets are noisy, chaotic environments that tend to drive us instinctively towards system 1 “survival mode” thinking so an already difficult situation becomes even harder.
Contrarian trading is not just attempting to do the opposite to the crowd. Sometimes the crowd is right and sometimes the opposite of wrong is just an opposite wrong.
Contrarian thinking is about using our conscious system 2 thinking to develop the ability to fearlessly step outside the consensus view, free ourselves from conformist thinking, see the bigger, often quite different picture and ultimately have trust and confidence in our own good judgement no matter what experts or others may say.