But think about the same game with 100 doors does your gut still tell you is a 50 50 toss up between your first pick and the one other remaining door?
It doesn’t matter how many doors we start with, when I’m asked to make that final choice, there are only 2 doors to choose from.
@VirtKitty This aspect was key for me to get the math right in my head. If you were trained in statistics & probability in the same way I was, we got used to the idea that each selection kicked off re-randomization. In the case of the game, that was against the rules. If the remaining doors were re-randomized, then it would be 50-50. But it’s not, and MH can’t (won’t?) pick the door with the prize.
Try it a few times with cards.
Seriously use more than 3, it will work.
This is proven math.
This was taught to me by a professional at school, it took me a while to get but it’s real.
So let me see if I have my head wrapped around this.
You’re saying the scenario is door 1 and door 2 closed and door 3 open OR door 1 and door 3 closed and door 2 open?? I’m clearly not good enough at math to understand how that’s valid. Must be something along the lines of viewing the event changes the outcome.
Peace out… and thanks for trying to explain it to me.
One more try?
I think @SamSam said it a few posts up. Imagine the scenario slightly differently. Monty never opens a door. He offers you the choice between the one door you already picked and the remaining two doors. Assuming he’s acting in good faith, what are your odds with Door 1 or Doors 2&3, combined?
Yes, I totally agree. That’s a 2/3 chance of being right to switch. But that’s having to make the choice BEFORE one of the doors has been eliminated.
Your contention is that, once there are only two doors remaining, you have 50% chance of getting right, even if you stick with your initial choice.
By the rules, Monty will always open all the doors but yours and one other, leaving two shut.
So, by your understanding, you always will have 50% chance of your first choice being right, no matter how many doors there were? That’s very impressive guessing!! But I’d call you cheating…
No, rather the chance of you getting it right on your first guess must have been 1/3rd and must remain 1/3rd.
If you were right on your initial guess and you swap, you lose. If you were wrong on your initial guess and you swap, you win. Which is more likely?
But you play the whole game each time.
Probability is the chance something will happen over multiple iterations.
It’s not just dividing by the number of options left at the last step of the game.
It’s the probabilities of winning the whole game, 2 out of 3 switching will win the game.
It doesn’t become 1 out of 2 mid way through the game.
The prize never moved. The door MH opens isn’t random - there is always an empty door available and he only picks an empty door. Opening one of the two remaining doors doesn’t change the prize’s location. That makes opening a door irrelevant in the odds.
Okay, after reading the Wikipedia article, it does make sense to switch.
Which is exactly what you do do. Then he opens one of the two without the prize. Which means your choice still is under those initial odds – 1 in 3. Since you know can eliminate the door that got opened, if you stay, it’s on those 1 in 3 odds. Switching brings you 2 in 3: the door you didn’t pick and the newly revealed losing door.
It’s counter-intuitive, but it works. Think of it as the two doors you didn’t choose merging into one location when Monty opens the door. Or, think of it as being abe to keep you choice or switch to the other two (not one of them, but both).
Okay then, I will grant that the odds are 2/3 when switching if you have to make the choice before the door is opened (or for calculating the odds for the whole game v. at the time of choice). But I’ve always seen the problem given as this (bolded part by me):
You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?”
That implies the choice is made after door 3 is eliminated. At least that’s the way I read it.
The thing is Monty’s choices differ depending on whether you have picked the correct door or not. If you have picked the incorrect door, there is only one door that he can open: the one that is NOT your choice and NOT the winner. If you have picked the correct door, he can pick EITHER of the other doors. This means that on average, he will have picked a particular, other door half as often when you have picked the winner, because in that case he has two different doors to choose from.
This is the point I was attempting to make above, where I said that you shouldn’t think of your first selection as a “choice”, but rather blocking off one of Monty’s doors.
In the (more likely) scenario where the door you block is a losing door, Monty’s only allowed action is to open the other losing door, leaving you with a winning door remaining.
Monty’s choices, were he ever to have made them, would have been based on how he was feeling that day.
But that is not how any probability works.
Probability is the actual likelihood of the thing happening.
If you wait until he says the magic words to make your decision then it won’t become more likely that you chose the right door in the first place.
You won’t win half the time. You need to actually win half the time for 50 50 odds, in real trials. Odds mean something they are not just on paper.
Staying will still loose 2 out if 3 games.
Nothing in your head can change that. The game is set at the first choice. It can’t be reset.
Edit:. I just want to make sure my tone isn’t mean or anything. I’m just repeating in different ways to see if it will click. Which is usually how hard concepts eventually click in my brain.
Monty is what people in my business like to call an “unreliable narrator”.
What I could never understand, watching this show in the 60s, is why anyone would prefer a radar-range oven to a goat.
To make that call, I would have to see the goat first.
Actually, I don’t know if in the 2 episodes I watched there was a goat. I do remember the radar range oven, because I can hear it in the announcer’s voice in my head. One of them had a lion cub, I think. Was there ever a goat?
From this interview:
But in 99 percent of the cases, we would offer them something after the show - a washer and dryer or a color TV or something, instead of that very valuable zonk, and they would take it. In 1 percent of the cases, they didn’t.
There was a time when a farmer won five calves and he wanted the calves. That cost me a fortune because when you rent them from the animal place, they’re expensive.
If they are told the whole scenario (of monty eliminating almost a million doors), then yes, they do pick the other door if they are told which door you picked. In this case they have the same knowledge as you do. If all they are given is a choice of two doors, without the other information, then from their standpoint it’s 50/50…but that’s because their knowledge is now incomplete compared to yours
