Proof of the Monty Hall Probability Puzzle
By Sebastian Lee

Created in response to this SmartKit page.

Number of Random Scenarios:
ScenarioFiguresStaySwitch

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Final Results# Correct3961
% Correct39%61%

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!