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Jennifer was selected to be on the popular TV game show "Whatchya Gonna Do?". As she jumped up and down in excitement, the host, Monty Barker, showed her three doors. "Now, Jennifer, behind one of these doors I have personally placed a brand new Jaguar XJS" "Eeeeeeee!", Jennifer squealed in delight. "But", continued MB, "behind the other two doors, there are goats! Select a door and the prize behind it is yours. Which shall it be: Door #1?". "Oh-oh-oh-oh", Jenny jumped. "Door #2?" "Ah-ah-ah-ah" or "Door #3?" "Um-um-um-um....well...ah...okay - TWO! I choose Door #2". The audience cheers. "Door #2. Okay Jennifer", says the host, "but before we show you what's behind Door #2, let me tell you that I am glad you did not pick Door #1, because behind Door #1 is..." The first door swings open to reveal a shabby looking goat. Jenny bounces around the stage in joy because she was thinking of picking that door. "Ah, but the name of the game is 'Watchya Gonna Do?'. You picked Door #2, but there is still Door #3. I'm going to offer you a choice: either you can keep the door you picked, Door #2, and if the car is behind it, I'll throw in a dinner for two at the Hoi Poloi Restaurant..." The audience gives a collective "oooooo" (who wouldn't want to eat at the exclusive Hoi Poloi?) "... or you can trade it for what's behind Door #3 - AND if it is the car, I'll throw in a year's supply of Platypus Wax! Tell us about the Platypus Wax, Joe" Joe's voiceover proceeds to tell us all about the wonders of using Platypus Wax on a Jaguar. As he drones on, Jennifer is nervously trying to decide. Until, MB spins around, points to her, and says (with the audience chorusing behind him) "What - Chya - Gon - Na - Do?"

Should Jennifer stay with her original choice (Door #2), switch to Door #3, or does it make any difference?

Assume the dinner at the High Poloi and the Platypus Wax are worth about the same dollar amount.

Aaaaaaand, before any of you wise-acres start, Jennifer lives in Toronto - the Jaguar is of far more use to her than the goat.

Spoiler for solution

Jennifer should switch. Contrary to what may seem intuitive, switching actually doubles her chances of winning the car. This problem is just a re-wording of what is known as the Monty Hall Problem. The key to understanding it is that the host knows the locations of the car and goats. His knowledge changes his actions and thus affects the odds. Here is a breakdown of all the possible scenarios that Jennifer faces and why Jennifer should switch:

Door #2 has goat B (probability 1:3) - MB shows goat A behind Door #1 (1:1) - the car is behind Door #3 (1:1) - switching wins the car - total chances (1:3 x 1:1 x 1:1 = 1:3)[/*:m:6a353]

Door #2 has the car (probability 1:3) - MB shows goat A behind Door #1 (1:2) - goat B is behind Door #3 (1:1) - not switching wins the car - total chances (1:3 x 1:2 x 1:1 = 1:6)[/*:m:6a353]

Door #2 has the car (probability 1:3) - MB shows goat B behind Door #1 (1:2) - goat A is behind Door #3 (1:1) - not switching wins the car - total chances (1:3 x 1:2 x 1:1 = 1:6)[/*:m:6a353]

There are (1:3 + 1:3 = 2:3) chances that switching will get Jennifer the car, and only (1:6 + 1:6 = 1:3) chances she would get the car by not switching. She should switch.

A more general presentation of the reasoning is this: At the start of the game, there is a 2:3 chance that Jennifer will pick a door with a goat behind it. If she does, the host will reveal the other other goat, and switching doors will get Jennifer the car. There is a 1:3 chance she will pick the car. The host will then reveal a goat. Switching would win Jennifer a good supply of Ch?vre (and the disdain of her neighbours). So, 2 out of 3 times switching gets the car. Simple - unintuitive, but simple.

Why does the host's knowledge change the odds. Because he does not randomly select a door to open - he always opens a door with a goat. By doing this he reduced the possible scenarios for Jennifer to the four listed above. If he randomly picked, then Jenny's chances, if the show progressed as presented, would be 50/50. However, there would also be a 1:3 chance that MB would open the wrong door and reveal the car's location (followed by a 1:1 chance that MB would be sacked and re-runs of McGyver would fill out the remainder of the season!)

I will refrain from answering this riddle as I encountered recently and after several hours of debating, I finally saw the light and understood the justification behind the correct answer.

I did want to give cpotting a round of applause for the great setup.

Its a very interesting puzzle. However I have a question in order to solve this, otherwise my answers dont make any sense.

Question: Since the question states

Should Jennifer stay with her original choice (Door #2), switch to Door #3, or does it make any difference?

Will he show her what is behind that remaining door, after she picks one? If Yes, then I think she should Stay with Door #2, Because it seems like the host is just playing with her mind. If No, then yeah it doesnt make any diffrence hoping that both other doors have cars, and since this is just a show, which ever sponsor gets lucky will pay the host

Will he show her what is behind that remaining door, after she picks one? If Yes, then I think she should Stay with Door #2, Because it seems like the host is just playing with her mind.

There's no need for him to reveal what's behind the remaining door if she decides to switch (other than to prove the show was honest about what was behind the doors). After she's seen what's behind two of the doors, what's behind the remaining door is apparent.

All that matters is that the host's policy is always to reveal a goat behind one of two doors that weren't picked and give the contestant the opportunity to switch. If the host always behaves in this manner, should you switch, not switch, or does it not make a difference?

Honestly, I am very interested in seeing your answer for this, cpotting.

The only possible hints that might lead to the answer is using the name "Hoi Poloi" for a restraunt.

After looking into it the term "hoi polloi" is used to refference the general public in a derogatory sense. As if to catogorize the group as "commoners."

In that sense, commoners would not own Jaguars, so if this was some sort of clue then I would say that door #3 would be the right pick.

But other than that, I would say it doesnt make a difference. If it is not stated that the contestants are given any valid hints, then no matter what anybody says it wouldn't matter which door she picked because there is a 50% chance that it is behind either of the doors and without any hints she has no way of knowing which one it is.

Thats my go for an answer. But I do want to find out what you have to say about it. Im dying to know your answer.

hmmm... So if she switches, there are more chances of her winning the car?

I'm sorry but i dont think this answer makes any sense....

Door #2 has goat A (probability 1:3) - MB shows goat B behind Door #1 (1:1) - the car is behind Door #3 (1:1) - switching wins the car - total chances for this scenario (1:3 x 1:1 x 1:1 = 1:3)

Door #2 has goat B (probability 1:3) - MB shows goat A behind Door #1 (1:1) - the car is behind Door #3 (1:1) - switching wins the car - total chances (1:3 x 1:1 x 1:1 = 1:3)

Door #3 has the car (probablility 1:3) - MB shows goat A behind Door #1 (1:2) - goat B is behind Door #3 (1:1) - not switching wins the car - total chances (1:3 x 1:2 x 1:1 = 1:6)

Door #3 has the car (probablility 1:3) - MB shows goat B behind Door #1 (1:2) - goat A is behind Door #3 (1:1) - not switching wins the car - total chances (1:3 x 1:2 x 1:1 = 1:6)

you never show possibility of Door #2 having the car.

and overall I dont think its a math question, in my opinion its more of a Logic or WordGame puzzle, meaning you should make a decision depending on Hosts reactions and your own intuition. What if the car is behind Door #2, then your whole math will be wrong, and she is left out with out a car.....

i know that i'm not 100% right. however i think thats a legit solution to that puzzle.

you never show possibility of Door #2 having the car.

If there is one thing that I have learned over the last few days, it is that posing problems over an internet forum is easy - editing them is harder than getting a cat to take a bath!.

You are right - the text should have read:

Door #2 has goat A (probability 1:3) - MB shows goat B behind Door #1 (1:1) - the car is behind Door #3 (1:1) - switching wins the car - total chances for this scenario (1:3 x 1:1 x 1:1 = 1:3)

Door #2 has goat B (probability 1:3) - MB shows goat A behind Door #1 (1:1) - the car is behind Door #3 (1:1) - switching wins the car - total chances (1:3 x 1:1 x 1:1 = 1:3)

Door #2 has the car (probablility 1:3) - MB shows goat A behind Door #1 (1:2) - goat B is behind Door #3 (1:1) - not switching wins the car - total chances (1:3 x 1:2 x 1:1 = 1:6)

Door #2 has the car (probablility 1:3) - MB shows goat B behind Door #1 (1:2) - goat A is behind Door #3 (1:1) - not switching wins the car - total chances (1:3 x 1:2 x 1:1 = 1:6)

Sorry for the confusion - I will edit the original and hopefully it will make more sense. All these errors... it's embarrassing