This post presumes you are familiar with the Monty Hall Problem and the associated post/article. If not, first read the Monty Hall Problem posting and then come back to this. Now we can begin…
Lock The Gates!
While ruminating on Monty Hall I came up with this Gedankenexperiment. It’s doubtful that what is here is unique/new but here it is anyway. Enjoy.
First, let’s restate the problem and refresh our memory.
You’re on the Let’s Make A Deal game show. You’re given the choice of three doors: Behind one door is a shiny new car!; behind the other 2 doors are goats (which you don’t want!). You pick a door, say No. 1, and the host (Monty Hall), who knows what’s behind the doors, opens another door, say No. 3, which has a goat. Monty then says to you, “Do you want to switch and instead pick door No. 2?”
What should you do?
1. Should you stick with your original choice of door No. 1?
2. Should you switch to door No. 2?
3. Or… as a whole lotta people believe, it wouldn’t matter whether you stick with door #1 or whether you switch to door #2. All you know is that one door has the prize and one doesn’t so you would have a 50/50 chance of winning either way. This is the part of the problem we will now focus on.
Let’s think about this…
When we initially picked door#1 our odds of winning the car were 1/3 (33.3%). Now, when Monty shows us a “goat” behind door#3, all of a sudden we think our odds of winning increased to 1/2 (50%) even if we stick with Door#1. But think about it… Did Monty really do anything that would change our initial odds of winning?
I like this one… What if Monty shows us the goat behind door#3 and then closes it, and then, 5 seconds later a meteor falls from the sky, hits us in the head, and as a result we forget what was behind the door!!….. And then we are given the chance to switch from our initial door#1 to door#2. If we stick with our original choice of door#1, what are our odds of winning the new car? Are they still the original 1/3 or have they increased to 1/2 ? Did the odds of the car being behind our original choice of door#1 go from 1/3 to 1/2 when we realized a goat was behind door#3? And did the odds then revert back to 1/3 when we forgot what happened?
Or how about this. What if we have a personality disorder that makes us always stick with our first choice no matter what. If Monty shows us the goat behind door#3 are the odds of the prize being behind door#1 and door#2 the same (50/50)? If we will always stick with our original choice will our odds of winning always be 1/3? That would seem to conflict with the notion that “All you know is that one of the 2 remaining doors has the prize and one doesn’t so you would have a 50/50 chance of winning either way.”
As we can now see, there are too many holes in choice #3 above. Or in other words, it does matter whether we stick with our original choice vs whether we switch to door#2 !!!
Open The Gates!