Let's Make A Deal

[LowRyter]

Since the V twin crank intervals controversy rages, why not start a new logic problem? 

This is an oldie but a goody.  I am sure many remember it.  Back in the '70s there was a game show known as "Let's Make a Deal".  Usually the ending was huge prize like a car or something.  The presenter, Monty Hall, would show 3 doors.  Behind one door was a new car, in the other two, only gag gifts.  Monty would let the contestant pick a door.  Then he would open another door to reveal a gag gift, he would then give the contestant the option of keeping his original choice or switch choice to the remaining door.

So, should the contestant keep his original choice, switch, or it really doesn't matter.







(hint: at no time are the odds 50/50 or even)

[twowings]

I want lovely Carol Merrill...behind the curtain...

[LowRyter]

you'll get the goat.   

[Two Checks]

Dont get zonked!
Furs came from Dicker and Dicker of Beverly Hills.
And many gifts came from Spiegel, Chicago, IL 60609.

[Scud]

The contestant should choose the remaining door.

Odds - 1 in 3 of choosing correctly the first time. (so 2 out of 3 times the good prize is behind one of the other two doors.)

Monty must open a door that does not have the good prize, so if the prize was not originally selected he does not (technically) have a choice.

Therefore, after Monty opens a door the odds are 1 in 3 that you picked prize originally and 2 in 3 that the prize is behind the remaining door.

However, the contestant's brain squirted some chemicals when he/she made the original choice. The contestants wants to *feel* right and mostly makes the illogical choice... or they never took probability and statistics and lack the ability to analyze the situation anyway.

[LowRyter]

too easy peasy.

Scud, drive off in your new Celica. 








I will admit I never got the logic of it until I read an example with 10 doors and considered the odds after 8 being opened.

[Scud]

Thanks. The Celicas of that vintage were great cars. Then they morphed into the Celica Supras that were surprisingly sporty (not quite like the original Datsun Z-cars, but similar to the sport-bloated 300ZX). I wonder if those are becoming collectible?

[Demar]

Where was the strangest place you ever made whoopie?  Oh wait..... that was the other game show.

[wymple]

Where was the strangest place you ever made whoopie?  Oh wait..... that was the other game show.

And it was a permanent classic. I still laugh at his answer and her reaction.

[guzzisteve]

And behind door #2 is a Degree Wheel made w/spiral in red paint so you can sit and watch THEN enter the Twilight Zone------

[Scud]

And it was a permanent classic. I still laugh at his answer and her reaction.

https://www.youtube.com/watch?v=4XM5hbS7GlU

[Kent in Upstate NY]

http://math.ucsd.edu/~crypto/cgi-bin/MontyKnows/monty2?0+12429

[MotoChuck250]

http://math.ucsd.edu/~crypto/cgi-bin/MontyKnows/monty2?0+12429

Yeah, I have heard that explanation before.  I think even the CarTalk guys have covered it on their radio show.   The explanation is not immediately intuitive but does make sense when you think about it for a while.

[LowRyter]

https://giphy.com/gifs/InKZtuwR1NXtC/html5

[Two Checks]

Whoopie? Now my mind needs eye bleach.

[Moto]

It does depend on how Monte is allowed to behave. If he only offers a second chance when your first answer was correct, you should of course not change curtains. The real Monte Hall was interviewed about this and explained that he did not in fact follow any particular rule, so the simple analysis doesn't pertain to the real show.

The clearest analysis I've seen of the abstract problem in the puzzle is this one, from a Decision Sciences professor:

*************************************************************************************

As we drive home in my car, which happens to be a Cadillac, I think to myself that I must develop a decision tree to work the problem. I'll assume that the game show Cadillac is behind Door #1. Then suppose I guess Door #1. This is the first branch of my tree. The host will not open Door #1, but he may open Door #2 or #3. Suppose he opens #2, and I switch to Door #3, then I lose. Not good.

"Red light!" Dick says, bringing me back to the other reality.

I jam on the brakes, the anti-lock system is activated, and the seatbelts keep us from colliding with the dashboard. Then we settle down, continue on our way, and I return to my world of math.

I visualize the second branch of the tree. I guess Door #2, and the host reveals #3. What if I switch to #1 and win? Much better. Now suppose I guess #3, the third branch of the tree. The host opens #2, I guess #1 and win. I can't believe it. Two of the three branches of the tree lead to winning. So the probability is 2/3 if I switch. Of course, I can't believe such nonsense, so I keep mum about it.

From: https://web.archive.org/web/20140413131827/http://www.decisionsciences.org/DecisionLine/Vol30/30_1/vazs30_1.pdf

*************************************************************************************

His analysis is correct. Interestingly, he also recounts how the famous mathematician Paul Erdos was visiting him at the time and could not be convinced of the answer until a simulation was done. (I believe Erdos was the most productive mathematician of the 20th century, at least as measured by numbers of coauthored papers; as a result mathematicians and scientists now sport "Erdos numbers" indicating their Kevin-Bacon-like degrees of separation from Erdos by chains of co-authorship.)