A probability question that just doesn't seem to make sense?!?

Question

You have three doors. Behind one of those three doors you have a Porsche. The other two contain nothing. Say you choose door number 1. Afterward, god (yes, god) opens door number 3 and tells you there is nothing in there. You are then left with only door number 1 and 2. By switching your first choice of number 1 to door number 2, you have a 33.3% higher chance that you will win the car. Could someone explain why?

a random side note, this was mentioned in the movie 21, at the fairly beginning section.

Thanks

Answer 1

This sounds like the "Monty Hall Problem."

Three doors; one has a prize.  If you pick the door with the prize, you get the prize.  You pick a door, Monty Hall opens a different door that does not contain the prize and allows you the right to switch you choice to the remaining unopened door.  Should you switch?

The answer is that it is advantageous to switch: your probability of winning the prize is 2/3 if you do so.  The probability that your first guess is wrong is obviously 2/3, and switching doors will gain you the prize if and only if your first guess was wrong.

The solution assumes that Monty knows which door contains the prize and that his decision to offer you the switch is independent of whether you originally chose the correct door.  These assumptions are implicitly made because they were the way the game was conducted by Monty Hall on "Let's Make A Deal" -- if you don't consider these assumptions to be natural then your conclusions may be different.

Answer 2

yes, the reason for this logic is that the director or whoever wrote that portion of the film doesn't have very good math skills.

Answer 3

Oh dear geezus -- the 100th time this month that the "Monty Hall" problem appears.

[Note, however, there ie s 1/3 probability of getting the car if you stick with your original pick. There is a 2/3 probability that you will get the car if you switch. You double your chances -- you have a 100% higher chance that you will win the car.]

OK -- so you pick a door. You know there is a 1/3 probability of winning. And you know that there is a 2/3 probability that it is behind one of the doors that you didn't pick, right? And you also know that it is certain that at least one of the doors that you didn't pick does NOT have the car behind it, right?

If god offered to let you trade the door you picked for BOTH of the ones that you didn't pick, you would trade, right? Because there is a 2/3 probability that it is behind one of the unpicked doors, right? And you know that at least one of those two doors does NOT have a car, right?

OK -- what if god showed you exactly what you already knew -- that there wasn't a car behind one of those two? Isn't there still a 2/3 probability that that the car is behind one of the doors that you didn't pick? You were already certain that one of them didn't have the car. Have you learned anything that you didn't already know? Isn't there still a 1/3 probability that your original door hides the car?

The reason that the probabilities don't change is that god doesn't open the door at random. He opens one (perhaps the only one) of the doors that he knows doesn't have the car. The key word is RANDOM RANDOM RANDOM. Probability only applies where there is randomness.

If you still don't believe, instead of 3 doors, let's say there are 1000 doors. You pick one; god looks behind the remaining doors and opens 998 of them. Which door would you rather have? The door that you picked blind at random, or the one that god (who knows what's behind every door) decided not to open?

Answer 4

The reason is that to begin with you only have a 33.3% chance of getting the right door in the first place. Reason being you have 3 choices, and 100 divided by 3 is 33.3%. Therefore each individual door has a 33.3% chance of being THE ONE... You open door number one and you've got nothing so you get to choose another door, as has been afore mentioned each door has a 33.3% chance of being the door with the car, therefore opening a second door gives you another 33.3% chance equaling 66.6% chance, so in general you do have a 33.3% higher chance, it's just being said in a funny way.  So God or anyone else opening door number three doesn't matter. Hope that helps!