Monty Hall Problem: Understanding the Difference

  • Thread starter aaaa202
  • Start date
  • Tags
    Monty hall
In summary: However, he hasn't given you any new information about the door you originally picked.In summary, the conversation discusses the famous problem of switching doors in a game show. The difference between picking a door and having the host reveal a goat, versus not picking a door and the host revealing a goat, is that in the former scenario, the host must open another door, giving you more information about the remaining doors. This is illustrated by the example of having 1,000 doors, one of which has a car and the rest have goats. If the host opens 998 doors with goats, the probability of the car being behind the unopened door increases to 999/1000. Essentially, the host's action gives you more information and
  • #1
aaaa202
1,169
2
I suppose you all know this famous problem. It is pretty clear to me why switching doors is beneficial, but I'm however unable to counter this argument from my friend:

What is the difference between having picked a door and then the host revealing a goat, compared to not having picked one and then the host revealing the goat. Clearly there is difference but can someone elaborate in understandable terms because I tend to get very confused.
 
Physics news on Phys.org
  • #2
The difference is that when you pick the door, the host must open ANOTHER door. Therefore if you had the goat in your door (which is 2/3 likely) then the door left closed will have the car 2/3 likely. This gives you information.

If you hadn't picked a door, then the host could choose any of the doors with goats at random, thus not giving any other information about the closed doors.
 
  • #3
Boorglar said:
This gives you information.
To hammer this point home, imagine if there were 1,000 doors, one of which hides a car and each of the the remaining 999 doors hides a goat. You pick randomly a door. The probability you picked the right door is a paltry 1/1000. Monty then opens 998 doors, each of which shows a goat. Do you switch? Of course you do. The probability that the car is behind the unchosen, unopened door is 999/1000.

Another way to look at it: Monty has just given you information, a whole lot of information in this case.
 

1. What is the Monty Hall Problem?

The Monty Hall Problem is a probability puzzle named after the host of the game show Let's Make a Deal. It is based on a game in which a contestant is presented with three doors, one of which contains a prize. After the contestant chooses a door, the host (Monty Hall) opens one of the remaining doors, revealing no prize. The contestant is then given the option to switch their choice to the other unopened door.

2. What is the difference between the two strategies in the Monty Hall Problem?

The two strategies in the Monty Hall Problem are "stay" and "switch". In the "stay" strategy, the contestant chooses a door and sticks with their initial choice, while in the "switch" strategy, the contestant changes their choice to the other unopened door after the host reveals an empty door.

3. Which strategy is more likely to lead to winning the prize in the Monty Hall Problem?

The "switch" strategy is more likely to lead to winning the prize in the Monty Hall Problem. This is because when the contestant initially chooses a door, they have a 1/3 chance of choosing the correct door. When the host reveals an empty door, the remaining unopened door has a 2/3 chance of being the correct door. Therefore, by switching, the contestant increases their chances of winning from 1/3 to 2/3.

4. Why does the Monty Hall Problem seem counterintuitive?

The Monty Hall Problem can seem counterintuitive because our brains tend to focus on the two doors that are initially presented as having an equal chance of containing the prize. However, once one of these doors is revealed to be empty, the odds shift and the remaining door has a higher probability of being the winning door. This concept can be difficult for our brains to grasp, leading to the counterintuitive nature of the problem.

5. Is there a mathematical explanation for why the "switch" strategy is more likely to win in the Monty Hall Problem?

Yes, there is a mathematical explanation for why the "switch" strategy is more likely to win in the Monty Hall Problem. The probability of initially choosing the correct door is 1/3, while the probability of initially choosing the incorrect door is 2/3. When one of the incorrect doors is revealed, the remaining door still has a 2/3 chance of being the correct door. The principle of conditional probability can be used to show that switching to this remaining door increases the chances of winning to 2/3, making it the more favorable strategy.

Similar threads

I Variation of Monty Hall problem
    • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
B Monty Hall Paradox: Why Does It Matter What Host Knows?
    • General Math
Replies
12
Views
1K
B Should You Stick or Switch? A Modified Monty Hall Problem
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
Comp Sci Monty Hall Game Show Problem in Javascript
    • Engineering and Comp Sci Homework Help
Replies
15
Views
1K
MHB Monty Hall Problem with 4 doors
    • General Math
Replies
1
Views
1K
I Does Monty Fall change the odds in the Monty Hall problem?
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
913
Monty Hall Problem: Probability Calculation for Switching Doors
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
B The Monty Hall paradox/conundrum
    • Set Theory, Logic, Probability, Statistics
7
Replies
212
Views
11K
I Monty Hall Problem -- Questions
    • General Math
Replies
30
Views
3K
Monty Hall: How Can the Host's Intentions Affect the Outcome of the Game?
    • Set Theory, Logic, Probability, Statistics
Replies
21
Views
13K

Hot Threads

Recent Insights

Back
Top