But the chance you picked correctly goes up to 50% after a door is opened.
No it doesn't. Why would it? You have a million possible choices, ONE of which is correct. If you picked that one, then switching will result in loss. If you picked ANYTHING else, then switching will result in victory.
If the second choice was 50/50 then that would mean you had a 50% chance of being right anyway. If you don't change your choice why should your chance of victory change?
Statistics is weird, but very fun once you've figured it out. I actually thought about becoming an actuary for a while because of how interesting they are.
Hey, I know. How about we actually play out this scenario a bunch of times via pm, and see what happens? We'll do it with 1 million doors. It should still be 50/50, right?
Nope. The trick is, you are essentially betting against yourself if you switch. Think about it, it you pick the wrong door(which is almost guaranteed) then all incorrect choices are eliminated. Only if you picked the right door to start with(highly unlikely) will switching mean you lose.
Okay, let's start from there.
100 doors, pick the wrong one. Host eliminates all other doors but the right one. You are now allowed to pick between the right door and the wrong door. 1/2.
100 doors, pick the right one. Host eliminates all doors except for yours and one empty door. You are now allowed to pick between the right door and the wrong door. 1/2.
Explain to me how this is wrong. And yes, I would very much like to see the results of a statistical program working out this problem.
Yes... but the first situation, in which the other door is correct, is 99 times more likely than the second. The trick to statistics is this: it doesn't matter how many possible states there are, it matters how many ways there are to reach each state.