Proof of the Monty Hall Probability Puzzle
By Sebastian Lee

Created in response to this SmartKit page.

Number of Random Scenarios:
ScenarioFiguresStaySwitch

#1Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#2Correct door #1StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#3Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#4Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#5Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#6Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#7Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#8Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#9Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#10Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#11Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#12Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#13Correct door #2StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#14Correct door #2StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#15Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#16Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#17Correct door #1StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#18Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#19Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#20Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#21Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#22Correct door #2StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#23Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#24Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#25Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#26Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#27Correct door #2StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#28Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#29Correct door #2StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#30Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#31Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#32Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#33Correct door #2StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#34Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#35Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#36Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#37Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#38Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#39Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#40Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#41Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#42Correct door #2StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#43Correct door #2StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#44Correct door #2StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#45Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#46Correct door #2StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#47Correct door #2StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#48Correct door #2StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#49Correct door #1StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#50Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#51Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#52Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#53Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#54Correct door #2StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#55Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#56Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#57Correct door #1StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#58Correct door #1StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#59Correct door #2StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#60Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#61Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#62Correct door #2StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#63Correct door #2StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#64Correct door #2StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#65Correct door #2StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#66Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#67Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#68Correct door #2StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#69Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#70Correct door #2StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#71Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#72Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#73Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#74Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#75Correct door #1StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#76Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#77Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#78Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#79Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#80Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#81Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#82Correct door #1StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#83Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#84Correct door #2StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#85Correct door #1StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#86Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#87Correct door #1StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#88Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

#89Correct door #1StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#90Correct door #2StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#91Correct door #2StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#92Correct door #3StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#93Correct door #1StaySwitch
Player chooses #1
Host reveals #2
Can switch to door #3

#94Correct door #3StaySwitch
Player chooses #3
Host reveals #1
Can switch to door #2

#95Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#96Correct door #3StaySwitch
Player chooses #3
Host reveals #2
Can switch to door #1

#97Correct door #1StaySwitch
Player chooses #1
Host reveals #3
Can switch to door #2

#98Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#99Correct door #3StaySwitch
Player chooses #2
Host reveals #1
Can switch to door #3

#100Correct door #1StaySwitch
Player chooses #2
Host reveals #3
Can switch to door #1

Final Results# Correct3664
% Correct36%64%

My explanation:

The fact that the host reveals one of the wrong answers AFTER you choose your initial answer does something really cool!

I think the easiest way to think of it is this:
1.) In order to win by switching your answer, all you have to do is choose a WRONG answer initially.
2.) You have a 2/3 chance of choosing a wrong answer initially!
3.) You have a 2/3 chance of choosing the right door by switching!

Another way of looking at it is this:
1.) The door you initially choose has a 1/3 chance of being correct. Each of the other two doors still have a 1/3 chance of being correct at this point.
2.) However, the host REVEALS one of these two other doors as wrong, meaning that door now has a 0/3 chance of being correct!
3.) So now there is 2/3 of a chance needing to be assigned to a door still… The last remaining door that you would switch to is the door that now has that 2/3 chance of being correct!