Posted by: thealienist | March 5, 2010

Pigeons and the “Monty Hall” Problem

The Set-Up

Let’s say that you have done particularly well on “Let’s Make a Deal” (an old game show hosted by Monty Hall and a remade game show hosted by Wayne Brady) and are now in the “Big Deal.”  You are shown three curtains and are allowed to pick one.  Let’s say that in this “Big Deal” one curtain conceals a car and the other two have “zonks” (a goat and a year’s supply of toothpicks).

You pick one curtain (let’s say curtain two).  Monty says, “Let’s see what you didn’t win.  Show me curtain number one!”  The curtain is lifted, revealing the goat.  You know that no matter which curtain you had chosen, Monty would have revealed one of the “zonks” thus leaving two curtains — your chosen one and another one.

The Question

Now Monty asks you if you want to keep the curtain you have chosen or if you want to switch to the other curtain.  What do you do?  Does it change your odds of winning?  The answer is below.

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The Answer

You should switch to the other curtain.  This doubles your chance of winning the car.

The Rationale

Consider the four following options:

1.  Your first pick selects the curtain hiding the car (random probability 1 in 3) and you decide to stick with your choice.

Monty has two curtains to choose from and can open either of them.  You stay with your curtain and win a car.

2.  Your first pick selects a “zonk” (probability 2 in 3) and you decide to stick with your choice.

Monty has only one curtain he can open and reveals the only remaining “zonk.”  You stay with your choice and win a goat or a year’s supply of toothpicks.

3.  Your first pick selects the curtain hiding the car (random probability 1 in 3) and you choose to switch.

Monty has two curtains to choose from and can open either of them.  You decide to change to the other remaining curtain and win a year’s supply of toothpicks.

4.  Your first pick selects a curtain hiding a “zonk” (random probability 2 in 3) and you choose to switch.

Monty has only one curtain he can open.  He reveals the only remaining “zonk.”  You decide to change to the other remaining curtain and win a car.

As the above shows, sticking with your first choice gives you a 1 in 3 chance of winning the car.  Changing your choice gives you a 2 in 3 chance of winning the car.  It’s foolish not to switch when given the chance.

The Surprise

Humans do not generally do well on the “Monty Hall” problem.  They tend to believe that switching choices after the first curtain is removed makes no difference in the odds of winning.  surprisingly, pigeons do much better.  It seems that they learn quickly from the actual pay-offs and are not confused by the sometimes confusing facts of conditional probability.  Oddly enough, young children also appear to do better than those who have progressed farther in their education.

Sometimes it pays to look at life like a “bird-brain.”

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