Monty Hall Paradox

Cockney Mackem

Cockney Mackem

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Bit of light thinking for a Sunday morning. This logic problem drove people crazy with how counterintuitive it is.

Named after American TV host Monty Hall, the scenario involves a contestant who is shown three doors, 1 2 and 3. Behind one door is the prize: a car, behind the other two doors are goats. The contestant has no clue where the prize is and has to guess. The host knows what's behind each door.

The contestant chooses door 1. The host then opens door 3 to reveal a goat, and asks if the contestant wants to change their choice. The contestant now knows the car is either behind door 1, their current choice, or door 2.

The bit that drives people up the wall is that the probability of winning the car if you switch to door 2 is 2/3. Why?
 


Cockney Mackem said:
Bit of light thinking for a Sunday morning. This logic problem drove people crazy with how counterintuitive it is.

Named after American TV host Monty Hall, the scenario involves a contestant who is shown three doors, 1 2 and 3. Behind one door is the prize: a car, behind the other two doors are goats. The contestant has no clue where the prize is and has to guess. The host knows what's behind each door.

The contestant chooses door 1. The host then opens door 3 to reveal a goat, and asks if the contestant wants to change their choice. The contestant now knows the car is either behind door 1, their current choice, or door 2.

The bit that drives people up the wall is that the probability of winning the car if you switch to door 2 is 2/3. Why?
Click to expand...
Because he is never going to open the door with the car in
 
Because you’ve already eliminated a wrong choice. Simple really.

The odds of you getting your first guess right is 1:3.

The odds of your second guess being right if you switch is 2:3 as you are in effect having two guesses.
 
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This did drive me mad for a good while until it finally clicked. The missus thought I was in a mood with her when we were out for a walk and I barely said a word. I was too busy trying to figure it out.

Can’t remember the exact figures but, basically, your first choice you’re choosing between three options. Second time round you choosing between two (if you switch). Your odds are therefore better.
 
Cockney Mackem said:
Bit of light thinking for a Sunday morning. This logic problem drove people crazy with how counterintuitive it is.

Named after American TV host Monty Hall, the scenario involves a contestant who is shown three doors, 1 2 and 3. Behind one door is the prize: a car, behind the other two doors are goats. The contestant has no clue where the prize is and has to guess. The host knows what's behind each door.

The contestant chooses door 1. The host then opens door 3 to reveal a goat, and asks if the contestant wants to change their choice. The contestant now knows the car is either behind door 1, their current choice, or door 2.

The bit that drives people up the wall is that the probability of winning the car if you switch to door 2 is 2/3. Why?
Click to expand...

By always switching your probability of winning is 2/3 because your original probability of choosing the goat is 2/3.

the possibilities of the switching strategy are:
Pick goat 1, see goat 2, switch to car
Pick goat 2, see goat 1, switch to car
Pick car, see either goat 1/2, switch to remaining goat.

Therefore 2/3 chance of winning the car since your original probability is 2/3 goat 1/3 car.
 
Legend said:
By always switching your probability of winning is 2/3 because your original probability of choosing the goat is 2/3.

the possibilities of the switching strategy are:
Pick goat 1, see goat 2, switch to car
Pick goat 2, see goat 1, switch to car
Pick car, see either goat 1/2, switch to remaining goat.

Therefore 2/3 chance of winning the car since your original probability is 2/3 goat 1/3 car.
Click to expand...

What I don't understand is that after one door is eliminated you are left with two doors that might have the prize behind it. Why aren't the chances 50/50?
 
Yeah I love this problem as so many people can't get their head around it.

You have a 1/3 chance of picking correctly initially, and there's a 2/3 chance the car is somewhere else.
Monty opens a goat door.
Your 1/3 chance of initially picking correctly still stands, and it's still a 2/3 chance of the car being somewhere else; but now the somewhere else only has one door left, so there's a 2/3 chance that the car is behind the door you didn't pick.
 
Cockney Mackem said:
What I don't understand is that after one door is eliminated you are left with two doors that might have the prize behind it. Why aren't the chances 50/50?
Click to expand...

Because there are more ways of the “switch” being a car.

The only way your original choice is a car is if you picked the car with a probability of 1/3. So 1/3 of the time you’d switch to a goat.

There are two ways that your original choice is a goat (goat 1 and goat 2) with a probability of 2/3. So 2/3 of the time you’d switch to a car.
 
Used to keep me awake at night thinking of this one.

There was a massive thread on here years ago about some woman who meets an old friend and she has a kid with her and says something like one of my other kids isn't a boy and the maths surrounding the odds of what kids she has is mind boggling.
Edit: this is it

en.m.wikipedia.org

Boy or girl paradox - Wikipedia

en.m.wikipedia.org en.m.wikipedia.org
 
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From Wikipedia:

Standard assumptions[edit]​

By the standard assumptions, the probability of winning the car after switching is 2/3. This solution is due to the behavior of the host. Ambiguities in the Parade version do not explicitly define the protocol of the host. However, Marilyn vos Savant's solution[3] printed alongside Whitaker's question implies, and both Selvin[1] and Savant[5] explicitly define, the role of the host as follows:

  1. The host must always open a door that was not selected by the contestant.[9]
  2. The host must always open a door to reveal a goat and never the car.
  3. The host must always offer the chance to switch between the door chosen originally and the closed door remaining.

^These are important. It doesn't work otherwise.
 
Big Sharp Teeth said:
This did drive me mad for a good while until it finally clicked. The missus thought I was in a mood with her when we were out for a walk and I barely said a word. I was too busy trying to figure it out.

Can’t remember the exact figures but, basically, your first choice you’re choosing between three options. Second time round you choosing between two (if you switch). Your odds are therefore better.
Click to expand...
You need to get out m........
Oh you were out. In that case i don't know what your answer is.
 
It's certainly a weird one when you first learn about it, there's some good videos on YouTube that explain it well which show how it all makes logical sense.
 

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