Pielorinho said:
[/i]
THAT'S RIGHT!
Stop right there.
Now, we have one more step. I'm Monty, and you're the guesser. You know, since you just told me this, that I have a 2/3 chance of having the full egg in my hand. Now I give you a chance to change your mind, to choose any of the three eggs.
See, right there is the flaw in your reasoning. There aren't three eggs to choose from. You didn't get what I'm trying to say a bit later - that there are only two real choices now.
You can choose the egg on the table. What's the chance that that's the right egg?
well, there are two viable eggs, one of which is a winner. That's 1/2
You can choose BOTH eggs in my hand, and if EITHER of them is right, then we'll proceed to the endgame. You just told me what the chances were that I was still holding the correct egg--what was it you said? 2/3, right?
Okay, so let's say you choose "the eggs in your hands" over "the eggs on the table." 2/3 of the time, that'll be the right choice, right?
Wrong. There aren't three choices, and you don't have 2 of them anymore. You have one, out of two.
If you're right, then I'll ask you the endgame question: of the two eggs in my hand, which one is the right one: the smashed one, or the whole one?
I'm going to assume that you'll choose the whole one 100% of the time, since obviously the smashed on isn't right. That means 2/3 of the time, you'll win!
As for your playing the game and being right 50% of the time, that baffles me. Do me a favor, wouldja? Play the game 30 times. Each time, choose 1 as your first guess, and then switch to the other one as your second guess. Tell me what you end up with. (I just went back to the game to verify my results and got an error).
Daniel
Results attached.
You're right that there is information gained when the egg is smashed. That information is:
"This egg is not a winner".
You do NOT tell me that the odds of your egg being the winner are 2/3 - why? Because there aren't 3 eggs in the game anymore, and you don't have 2 eggs anymore. You have 1. Out of 2.
I don't know how to make it any more simple.
2 eggs. 1 winner. 1 choice. 50% chance.
When we started the game, each egg has an equal chance of being the winner. That doesn't change because one egg is removed.
The 1 in 100 or 1000000 game as described is a fundamentally different game.
Here's why.
When you smash the egg, you are removing one egg from the game. You are not reducing the choices to two, you are removing one egg. The fact that this reduces the choices to 2 only happens to be a coincidence of the starting number.
The only way these are the same game is if you reduce the number by one in each game. In which case, the chances of being right = 1/2 or 1/99.
In the game Monty Haul played on TV, each choice has an equal chance of being right. No matter how many choices, as long as there is only a reduction of 1 choice at a time, the odds are equal to the number of choices.
You are right that there is information gained if more than one choice is removed at once - because that is a fundamentally different game than the game we are talking about, where you remove one at a time.
basically, he says, "here are 99 more guesses, and all of them are right." He packages all the losing guesses together with the win, which multiplies your odds of winning by switching by 99 times."
It's now a game with 99 right choices and one wrong choice.
You'd be stupid NOT to switch, because Monty gave away the game.
Imagine you start with three choices, and Monty removes two! That's a lot of information!
However, if you were to play the game with 100 doors, and Monty just keeps removing one door at a time and letting you guess all the way down to two doors, the answer still won't change! Your chance of winning = 1/2.
Back to the original question:
Imagine it with four choices, for a more clear example.
Four Doors. I choose #2.
Monty removes #1.
What's left?
#2, #3, #4.
What should I do? switch? stay?
It doesn't matter. Each door has a 1/3 chance of winning! Three doors. One winner. One choice.
See how the game doesn't matter until the second round? No matter what, the first bit of information isn't information at all - it's meaningless! It's just for excitement and drama.
Imagine if they only started with two doors. No drama, just a simple coin toss. BOOR-RING!
But by performing the sleight of hand with the two choices, they turn a simple coin toss into good *ahem* TV.
jtb
