A Probabilities Problem

[Rocketman]

Actually, cognitive biases (i.e. ways in which our intuition fails us) have been the subject of extensive research, beginning with Tversky and Kahnemann in the late seventies.

 

[Goliath]

^That's good to know.

 

[Mr Awe]

Mr Awe -- the boxes are not symmetric though. If they were, say if you were approaching the game at the two box stage and had no clue which was initially chosen, then it would be symmetric, and it would be 50-50..

The asymmetry is this: One box is a blind choice of 1 in 1,000,000. The other box has been selectively whittled down from a larger set. That isn't random. It has been fail boxes that have been removed. Each fail box removed makes that set of 'others boxes' more and more lucky.

In the three box game, you know which two boxes has been selectively whittled down to one, and you know which box is the blindly chosen one.

I'm in the process of wrapping my brain around this. I'm starting to buy into the fact that, yes, it is a good thing to switch. The game table seems to prove this for the 3 door version. I think you're right though. Essentially you have 2 groups at the outset. I'll use the 3 door for simplicity. You choose one and that's one group with 33% odds of the prize. The other group has 2 and the odds are 66%. Now, the 2nd group is whittled down selectively.

Here's the key for my understanding. The odds for both groups don't change from the beginning to the end, only the number of choices within that group. So, you get the 66% group down to one door and yet the odds remain at 66%. That's why the table shows you winning 2 out of 3 times when you switch.

Same principles apply to the million door example. 2 groups. 1 in a million chance for your pick versus 999,999/1,000,000 chance for the other group of remaining doors. You remove selectively whittle that larger group down but 999,999 will always be in the the numerator. Just divide by the number remaining in that group to figure the odds of winning, but it'll always be higher than your 1 in a million pick.

I think I got it!

Mr Awe

 

[Goliath]

I'm in the process of wrapping my brain around this.

You have an admirably pliable brain, sir.

 

[Deckerd]

And yet you meet people who claim to be very intuitive or are regarded as being very intuitive. I often wonder if they just cherrypick their correct guesses, like looking at a prediction from a 'psychic' and only seeing the bits that match your experience.

 

[Goliath]

And yet you meet people who claim to be very intuitive or are regarded as being very intuitive. I often wonder if they just cherrypick their correct guesses, like looking at a prediction from a 'psychic' and only seeing the bits that match your experience.

I'm sure you're right--though I'm not sure that's the same type of 'intuition.'

Successful scam artists like psychics are very good at what they call "cold reading"--picking up both verbal and nonverbal cues.

And the more willing people are to believe them, the more they're likely to cherrypick the correct answers and ignore the rest, as you suggest.

 

[Jadzia]

I am very intuitive; it's my dominant mental faculty. But being intuitive doesn't mean being often correct. Intuition is a process rather than a credibility.

 

[Rocketman]

And yet you meet people who claim to be very intuitive or are regarded as being very intuitive. I often wonder if they just cherrypick their correct guesses, like looking at a prediction from a 'psychic' and only seeing the bits that match your experience.

It's called confirmation bias.

Intuition isn't completely useless, though. It's just that probability is one of those areas that refuses to yield to it. Having spent some time amongst math majors who admitted to often being similarly stumped by a subject that 'ought to be simple', I'm more and more convinced that our brains aren't very much suited to it.

 

[Deckerd]

Intuition isn't completely useless, though.

Wasn't it Crimson Tide where Gene Hackman gave a glorious example of intuition?

 

[Zulu Romeo]

I'm more of an educated-guess person myself. I wouldn't consider myself intuitive... unless my life depended on it.

 

[Goliath]

I am very intuitive; it's my dominant mental faculty. But being intuitive doesn't mean being often correct. Intuition is a process rather than a credibility.

I think everyone operates mostly on the basis of intuition, habit, and custom, pausing only to try to reason things out when we encounter something counterintuitive.

And as Rocketman suggests, this reasoning is mostly directed toward accommodating these anomalies within the framework of our preconceptions.

Though potentially highly useful, reasoning is in many ways the least reliable of all our mental faculties. If we had to rely on reason alone to get us through life, we'd probably starve to death.

 

[IndyJones]

I think I got it!

Mr Awe

You got it faster than a lot of people. Even after Marilyn posted the table that I copied above, people were still so convinced that she was in correct that they sent the following responses:

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 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

And my favorite:

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

I wonder if he stuck with the column long enough to realize his error!

 

[Mr Awe]

^^^ I wonder if any of those Ph.D.s retracted their comments and, hopefully, apologized!?!

 

[Scout101]

I'm hoping she published all of their letters, as well as credentials, and then the little chart showing the answer again. And then sent courtesy copies to their department heads at their schools...