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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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!