Math Is Fun Forum

NanaDee

Member

Registered: 2009-02-11

Posts: 8

The Monty Hall puzzle

I am new to the forum, so I decided to look around and landed on the puzzle section. Not sure I understand the answer to the goats behind the door. There is a 1/3 chance of the goat being behind the door selected. When a door is revealed to have one goat, that leaves the 2 remaining doors (one has a goat and one a prize - a car? I cant remember). Anyway, now you have a 1/2 chance of winning. Why would you switch? The other door remaining has the same probability as the one you selected. Did I miss something here? It reminds me of the lighted sign above the roullette tables in Vegas. If you watch them, you think the probabilities are changing (i.e. 7 red numbers have come up, the next one must be a black).

integer

Member

Registered: 2008-02-21

Posts: 79

Re: The Monty Hall puzzle

What would your chances be if Monty Hall opened the door showing the winning prize?
Would that change your probability of winning?

janetpc2008

Member

Registered: 2009-02-03

Posts: 7

Re: The Monty Hall puzzle

There is a 1/3 chance of the goat being behind the door selected. When a door is revealed to have one goat, that leaves the 2 remaining doors (one has a goat and one a prize - a car? I cant remember). Anyway, now you have a 1/2 chance of winning.

NanaDee - what you say here is wrong. When you picked the door you did, the odds were 1:3 that you chose the car. EVEN WHEN another door is revealed, your chances of already having the car are STILL 1/3. This is a fundamental truth, the showing of the goat does not change that, because Monty Hall KNOWS what is behind the doors, and he makes sure that he reveals a goat and not the car. If a random person in the audience picked another door, and revealed what was behind it, that would be different. In this new case, the chance of a car being revealed is 1/3, and if a car was NOT revealed then NOW your chances are 1/2 that the car is behind the door you chose.

But, back to the original question, here are the scenarios (there are 6 permutations):

One goat is black (B), one goat is white (W), and of course there is the car (C).
The possible distribution of objects (the first column is the one you picked):
C B W   you already have the car , B or W goat is shown, if you switch you lose 1/6
C W B   you already have the car , B or W goat is shown, if you switch you lose 1/6
B C W   you chose the black goat, white goat is shown, if you switch you win 1/6
W C B   you chose the white goat, black goat is shown, if you switch you win 1/6
B W C   you chose the black goat, white goat is shown, if you switch you win 1/6
W B C   you chose the white goat, black goat is shown, if you switch you win 1/6

NOTICE - that Monty Hall has to be very careful which door he opens. In 4 out of the 6 scenarios, if he opens the wrong door, the car will be revealed!

If you stick with your original choice, your odds of winning are 1/6+1/6 = 1/3
If you switch, then they are 1/6+1/6+1/6+1/6 = 2/3

chesler

Member

Registered: 2009-08-15

Posts: 2

Re: The Monty Hall puzzle

Old thread,  much older question. It's not solvable unless you know if Monty will always show you a goat, or will show you a goat only if you originally chose the car; also does he know where the car is (and if not, what happens if he reveals the car?)
Example: I have a dollar in one fist, choose left or right?  Are the odds 1/2-1/2 if I'm waving one fist at you begging you to pick it? (And are you my to grandchild whom I'm trying to give a dollar, or are you some guy in the bar?)  Are the odds 1/2-1/2 if you've observed in a sufficiently large number of trials that I always put the dollar in my right hand when I play this game? Are the odds 1/2-1/2 if you can see the dollar coming out of my right hand?

Juanita99

Member

Registered: 2009-08-21

Posts: 1

Re: The Monty Hall puzzle

Old thread,  much older question. It's not solvable unless you know if Monty will always show you a goat, or will show you a goat only if you originally chose the car; also does he know where the car is (and if not, what happens if he reveals the car?)
Example: I have a dollar in one fist, choose left or right?  Are the odds 1/2-1/2 if I'm waving one fist at you begging you to pick it? (And are you my to grandchild whom I'm trying to give a dollar, or are you some guy in the bar?)  Are the odds 1/2-1/2 if you've observed in a sufficiently large number of trials that I always put the dollar in my right hand when I play this game? Are the odds 1/2-1/2 if you can see the dollar coming out of my right hand?

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