The Monty Hall Problem

[12intheBox]

Our older members may remember this one but for some of you whipper snappers - indulge me a moment.

There are 3 doors, A, B, and C behind which are two goats and a car.

You get to pick a door (call it door A). You’re hoping for the car of course.

Monty Hall, the game show host, examines the other doors (B & C) and opens one (let’s call it B) to reveal a goat.

Here’s the game:

You are given a choice to keep what is behind door A or take what is behind door C.
Do you stick with door A (original guess) or switch to the unopened door C? Does it matter, mathematically - and if so - why?

 

[RamBall]

It depends on how I chose A. If I'm just guessing, keep it. If something in my head told me to pick A, I'm switching it for C. Because that fucker in my head is always wrong.

 

[shovelpass]

The odds are now 50/50. I'm assuming the host doesn't know which door the car is behind, but if he's looking at the other 2, he possibly doesn't think it's behind A. I would switch to door C.

 

[Dodgersrf]

Your odds just went up from 1/3 to 1/2.
The odds say switch doors.

 

[12intheBox]

The odds are now 50/50. I'm assuming the host doesn't know which door the car is behind, but if he's looking at the other 2, he possibly doesn't think it's behind A. I would switch to door C.

The host absolutely knows which door has the car.

 

[12intheBox]

Your odds just went up from 1/3 to 1/2.
The odds say switch doors.

Why switch if the odds are 1/2?

There are 2 doors left and you have 1 of them.

 

[ScotsRam]

Why switch if the odds are 1/2?

There are 2 doors left and you have 1 of them.

Because your original odds when you made your choice was 1 in 3. The removal of one of the options doesn't change that, it's still 1 in 3 that you made the right choice. If you switch, you've changed your odds to 1 in 2.

 

[12intheBox]

Because your original odds when you made your choice was 1 in 3. The removal of one of the options doesn't change that, it's still 1 in 3 that you made the right choice. If you switch, you've changed your odds to 1 in 2.

Kind of - although it seems that keeping your door 1 is also 50% at that point.

You have a 50/50 choice to keep 1 or switch to 3.

The best explanation I have ever found for this problem is explained in this video:


View: https://youtu.be/7u6kFlWZOWg?si=Yb81WckxPP6bBhbE

 

[Loyal]

Unless the game is rigged and the host knows which door hides the car, why would you switch? It's 50/50 at that point...,.,

BTW, my parents were on Lets Make a Deal and the selection process isn't what you might think. Costumed couples stood in a long line before filming and producers would walk down the line, picking those with costumes the producer liked. My parents were picked to participate, but they were on the fringe of where Monte Hall would choose contestants and so didn't really have chance at the prizes. Both Mom and Dad said that Monte Hall was a mean scumbag when the cameras were off and intimidated his staff (Carol Merril, etc...), but then that smile and fake good humor would shine for filming. Dad called him a mean drunk.

 

[12intheBox]

Unless the game is rigged and the host knows which door hides the car, why would you switch? It's 50/50 at that point...,.,

BTW, my parents were on Lets Make a Deal and the selection process isn't what you might think. Costumed couples stood in a long line before filming and producers would walk down the line, picking those with costumes the producer liked. My parents were picked to participate, but they were on the fringe of where Monte Hall would choose contestants and so didn't really have chance at the prizes. Both Mom and Dad said that Monte Hall was a mean scumbag when the cameras were off and intimidated his staff (Carol Merril, etc...), but then that smile and fake good humor would shine for filming. Dad called him a mean drunk.

The host does know - he has to. That doesn’t make it rigged - but part of the game is that you pick one door and then he eliminates one of the doors w the goat.

 

[12intheBox]

Btw - that is awesome Monte Hall context @Loyal

 

[TSFH Fan]

This site includes a model that lets you see that switching is the way to go:

Understanding the Monty Hall Problem – BetterExplained

Surprisingly, the odds aren’t 50-50. If you switch doors you’ll win 2/3 of the time!

Summary​

Here’s the key points to understanding the Monty Hall puzzle:

• Two choices are 50-50 when you know nothing about them
• Monty helps us by “filtering” the bad choices on the other side. It’s a choice of a random guess and the “Champ door” that’s the best on the other side.
• In general, more information means you re-evaluate your choices.

The fatal flaw in the Monty Hall paradox is not taking Monty’s filtering into account, thinking the chances are the same before and after. But the goal isn’t to understand this puzzle — it’s to realize how subsequent actions & information challenge previous decisions. Happy math.