The Monty Hall Problem

Ok, this is an explanation of The Monty Hall Problem, so I won't go into what it is just look it up, but here's the express explanation on why you have BETTER probability to win if you switch doors.


So let's go through all of the possibilities, now I know that there are two doors that Monty can choose from to open if on your first pick (when there are three doors,) your door has a car behind it.  So I'm going to split the possibilities of which door he can open into 18 possible outcomes.  In all of these outcomes, I will tell you what happens if you switch.  Remember, in a scenario where switching leads to victory, not switching leads to losing.

In our first set of outcomes, Monty can only open the goat door closest to the left, and still HE CAN'T OPEN YOUR DOOR, and HE CAN'T OPEN THE DOOR WITH THE CAR BEHIND IT.  So this example is how our outcomes are gonna work:  In this example I will highlight all the 1s in red, all the 2s in yellow and all the 3s in green

• Outcome set 1

1. If the car is behind door 1 and you pick door 1, he opens door 2 and lets you switch to door 3
2. If the car is behind door 2 and you pick door 2, he opens door 1 and lets you switch to door 3
3. If the car is behind door 3 and you pick door 3, he opens door 1 and lets you switch to door 2

In our second set of outcomes, Monty will always open the goat door closest to the right if the door that you've chosen has the car behind it.

• Outcome set 2:

1. If the car is behind door 1 and you pick door 1, he opens door 3 and lets you switch to door 2
2. If the car is behind door 2 and you pick door 2, he opens door 3 and lets you switch to door 1
3. If the car is behind door 3 and you pick door 3, he opens door 2 and lets you switch to door 1

So before I finally get into out set of outcomes, I'd like to point out that even in our example of scenarios in which you lose if you switch, we have a pattern that doesn't lead to a 50% chance of getting either goat.  Let's assume that you want the goat that's closest to the left, in outcome set 1, you have a 2/3 chance of getting that goat and a 1/3 chance of getting the other.  You have the opposite odds in outcome set 2.  This is important because it means that if we want each goat to have an equal chance of being the goat that we take home, each of the goats has to have a 1/3 chance of being the goat that we go home with if we switch and lose.

So finally here's our set of outcomes:

1. The car is behind door 1:
1. You pick door 1; and Monty opens door 2 and lets you switch ONLY to door 3, YOU SWITCH AND LOSE
2. You pick door 2, and IN THIS SCENARIO, MONTY HAS A 100% CHANCE OF OPENING DOOR 3 AND LETTING YOU SWITCH TO DOOR 1, YOU SWITCH AND WIN
3. You pick door 3, and IN THIS SCENARIO, MONTY HAS A 100% CHANCE OF OPENING DOOR 2 AND LETTING YOU SWITCH TO DOOR 1, YOU SWITCH AND WIN
2. The car is behind door 2:
1. You pick door 1; and IN THIS SCENARIO, MONTY HAS A 100% CHANCE OF OPENING DOOR 3 AND LETTING YOU SWITCH TO DOOR 2, YOU SWITCH AND WIN
2. You pick door 2; and Monty opens door 1 and let's you switch ONLY to door 3, YOU SWITCH AND LOSE
3. You pick door 3; and IN THIS SCENARIO, MONTY HAS A 100% CHANCE OF OPENING DOOR 2 AND LETTING YOU SWITCH TO DOOR 2, YOU SWITCH AND WIN
3. The car is behind door 3:
1. You pick door 1; and IN THIS SCENARIO, MONTY HAS A 100% CHANCE OF OPENING DOOR 2 AND LETTING YOU SWITCH TO DOOR 3; YOU SWITCH AND WIN
2. You pick door 2; and IN THIS SCENARIO, MONTY HAS A 100% CHANCE OF OPENING DOOR 1 AND LETTING YOU SWITCH TO DOOR 3; YOU SWITCH AND WIN
3. You pick door 3; and Monty opens door 1 and let's you switch ONLY to door 2; YOU SWITCH AND LOSE
4. Now before we go on, I'd like to remind everyone that in —each scenario in which switching causes you to lose,— there's a 50% chance of him opening the left most goat door and a 50% chance of him opening the right most goat door.  We have to repeat these results because —each of the scenarios in which the car is not behind your door— is actually two; —one scenario in which if there was a car behind your door, Monty would reveal the left most goat and let you switch to the right most goat,— and —a scenario in which if there was a car behind your door, Monty would reveal the right-most goat and let you twitch to the left-most goat.—  So in the next examples, Monty will now open the right-most goat door if the car is behind —the door that you pick out of 3.—   So now if: The car is behind door 1:
1. You pick door 1; and Monty opens door 3 and lets you switch ONLY to door 2; YOU SWITCH AND LOSE, I want to point out here that this is the first in a set of scenarios is which Monty the left most door to let you switch to and YOU STILL LOSE if you switch.
2. You pick door 2; and IN THIS SCENARIO MONTY HAS A 100% OF OPENING DOOR 3 AND LETTING YOU SWITCH TO DOOR 1; YOU SWITCH AND WIN, now some of you are saying that I'm making the odds uneven by listing the winning scenarios twice, but IT DOESN'T MATTER WHICH GOAT MONTY LETS ME SWITCH TO, NEITHER GOAT IS A CAR, so if I have to list —the scenarios in which I switch to a goat— twice, I have to listed —the scenarios in which I switch to a car— twice.
3. You pick door 3; and IN THIS SCENARIO, MONTY HAS A 100% CHANCE OF OPENING DOOR 2 AND LETTING YOU SWITCH TO DOOR 1; YOU SWITCH AND WIN
5. So let's just recap what we've learned so far, for —the scenario in which the car is behind door 1:— we've already established 6 possibilities; in our first set, Monty will reveal the left goat and let you switch to the right goat if you pick door 1.  In our second set, Monty will reveal the right goat and let you switch to the left goat if you pick door 1.  So it's a 2/6 chance that switching will cause you to lose, and because we have to double the other possibilities as well, it's 4/6 that switching will cause you to win.  2/6 equals 1/3 and 4/6 equals 2/3, the odds of winning if you switch, and that trend will continue when we go to what happens if: The car is behind door 2:
1. You pick door 1; and IN THIS SCENARIO MONTY HAS A 100% CHANCE OF OPENING DOOR 3 AND LETTING YOU SWITCH TO DOOR 2; YOU SWITCH AND WIN
2. You pick door 2; and Monty opens door 3 and lets you switch ONLY to door 1; YOU SWITCH AND LOSE
3. You pick door 3; and IN THIS SCENARIO MONTY HAS A 100& CHANCE OF OPENING DOOR 1 AND LETTING YOU SWITCH TO DOOR 2; YOU SWITCH AND WIN
6. The car is behind door 3:
1. You pick door 1; and IN THIS SCENARIO MONTY HAS A 100% CHANCE OF OPENING DOOR 2 AND LETTING YOU SWITCH TO DOOR 3; YOU SWITCH AND WIN
2. You pick door 2; and IN THIS SCENARIO MONTY HAS A 100% CHANCE OF OPENING DOOR 1 AND LETTING YOU SWITCH TO DOOR 3; YOU SWITCH AND WIN
3. You pick door 3: and Monty opens door 2 and lets you switch to door 2; YOU SWITCH AND LOSE

If you hit control+F on this page and type the words, "YOU SWITCH AND WIN," you will see 13 matches including this example, that's 12/18 possibilities, 12/18=2/3  In all 18 possibilities, —whether Monty wants to reveal the left goat when he has a choice or when he wants to reveal the right goat when he has a choice— doesn't matter because when Monty has a choice of which goat to reveal, you lose.  But there's only a 1/3 chance the Monty will have a choice of which goat to reveal and no matter which goat he reveals, if Monty had a choice of which goat to reveal and you switch, you lose.  There's a 2/3 chance that Monty can ONLY reveal 1 goat, and that's more likely and that probably happened without out knowing it so it doesn't matter that there are two doors after one is eliminated because there's a 2/3 chance that you've forced Monty to tell you which door the car is behind, and when you've forced Monty to reveal which door the car is behind, it's behind the door that you end up switching to if you decided to switch.  If you decide to switch, you're NOT choosing out of the two remaining doors that you didn't pick when there were 3, when you choose to switch, you're choosing to win the car if there's a goat behind the door you picked when all 3 doors were closed, and there's a 2/3 chance of picking —a door with a goat behind it— when all 3 doors are closed.  When you switch, you're turning a 2/3 chance of losing into a 2/3 chance of winning.