## Probabilities from movie 21

 Quote by scalpmaster So it is the info of which door first player picks rather than knowing which door host reveal that gives the edge from 1/2 to 2/3.
What gives the advantage is a combination of two factors.
1. The contestants initial chance of picking the right door is 1/3. Nobody will argue this.
2. The host then opens one of the other two doors, reducing that doors chance of holding the prize from 1/3 to 0/3, and shifting that probability to the remaining unpicked door -- as combined they had a 2/3 chance of holding the winner.

If a 2nd player comes in who knows only which door the host opened, he has a 1/2 chance of picking the right door from the remaining two. If he knows only what door the 1st contestant picked, his odds of picking the right door are also 1/2. If he knows both things, then he can pick the door that the 1st contestant should switch to, and get the same 2/3 odds.

 Quote by scalpmaster The new player may not be aware but from the audience standpoint, is there some kind of a hidden transient change in the probability the moment he happens to pick the opposite as the first player by himself (the 2 doors are still not open yet)?
I don't know what you mean here, but I described all possible scenarios above.

 Quote by scalpmaster Given a fixed time period,say 1 week, the stock market can go up,down or stay sideways. A hedge fund can long, short or use rangebound(options strategy) to bet for that week closing level. Hedge fund manager A opened long position the market on monday. On tuesday, there is some major news/economic data that can be interpreted either way so the market will breakout of the trading range to move in one unknown direction(3rd door is out). Hedge fund manager B comes along after the news. Should he (1) randomly pick one direction? (2) instead of analysing market direction, find out fund manager A's position and bet the opposite?
In this scenario, sure, short everything. However stock markets are not nearly so unpredictable, and news that comes out on monday is going to affect the market on monday as soon as the news breaks -- not the next day.

 Quote by justsomeguy What gives the advantage is a combination of two factors. 1. The contestants initial chance of picking the right door is 1/3. Nobody will argue this.
Wrong.Only applies to the first contestant.2nd player is shown 2 doors only.

 Quote by justsomeguy 2. The host then opens one of the other two doors, reducing that doors chance of holding the prize from 1/3 to 0/3, and shifting that probability to the remaining unpicked door -- as combined they had a 2/3 chance of holding the winner.
Correct, the critical question is does it matter whether the host had opened the SAME empty door knowingly or randomly(independently guess) from the other two doors?
If the host knows everything, it is guarranteed that the game will proceed to the 3rd stage(to switch or not) since he will always pick an empty door.
If the host just guess one of the other 2 doors, the game may stop at 2nd stage(host reveal door).
The question is whether the overall situation is affected if the host guess the same empty door and the game still proceeds to stage 3 as per normal? Mathematicians seem to think it is...but why in layman terms?
http://math.stackexchange.com/questi...l-problem?lq=1
 Quote by justsomeguy If a 2nd player comes in who knows only which door the host opened, he has a 1/2 chance of picking the right door from the remaining two. If he knows only what door the 1st contestant picked, his odds of picking the right door are also 1/2. If he knows both things, then he can pick the door that the 1st contestant should switch to, and get the same 2/3 odds.
2nd player probability depends on whether he knows how many other doors existed before he is shown the 2 doors game he is playing and 1st contestant's choice. If he does not know anything, from his perspective, it is always 50-50.

 Quote by scalpmaster 2nd player probability depends on whether he knows how many other doors existed before he is shown the 2 doors game he is playing and 1st contestant's choice. If he does not know anything, from his perspective, it is always 50-50.
Yes, "from his perspective." If door A is the 1st contestants choice, B is the hidden door with the goat, and door C is the remaining door, there is still a 2/3 chance that door C holds the prize.

The fact that the 2nd contestant doesn't know which door (A or C) the 1st contestant picked does not change the odds of a door holding the prize -- which is what the puzzle is all about -- it just changes the odds of the 2nd contestant picking the door with better odds.

Do you see the difference?

 Quote by justsomeguy Yes, "from his perspective." If door A is the 1st contestants choice, B is the hidden door with the goat, and door C is the remaining door, there is still a 2/3 chance that door C holds the prize. The fact that the 2nd contestant doesn't know which door (A or C) the 1st contestant picked does not change the odds of a door holding the prize -- which is what the puzzle is all about -- it just changes the odds of the 2nd contestant picking the door with better odds. Do you see the difference?
All the above is self explanatory when I said "from his perspective", but thats not my question which is "does the overall mathematical condition change if the host guess instead of knowing the same empty door to reveal?"
 Your question needs more rules specified as I said when you first asked it. If the host opens a random door there are two special cases you need to address in the rules. Specifically, what happens when the host opens a random door and reveals the prize. Can the contestant switch to that door? What if it's the contestants door?

 Quote by justsomeguy Your question needs more rules specified as I said when you first asked it.If the host opens a random door there are two special cases you need to address in the rules. Specifically, what happens when the host opens a random door and reveals the prize. Can the contestant switch to that door? What if it's the contestants door?
Then there is simply no stage 3 for player 1 and no 2nd player.Game over.