Would you switch?
You are presented with three doors, one has one million dollars behind it, two have a penny. After making your choose, one of the doors with a penny behind it will be opened and you have the option to switch doors. Do you switch?
This is an interesting puzzle that has baffled many people, including some extremely smart people. I came across the puzzle recently and I find it interesting how it either makes total sense to you, or you believe it is complete bullshit.
The fact is, you have double the chances of winning the 1 million dollars if you switch doors once you are told one of the doors that has a penny behind it. You have a two out of three chance to win the million dollars if you switch doors every time while only having a one out of three chance if you stick with your first door.
One way to look at it is like this, you have a one in three chance of picking the money on your first try. You have two out of three chances of picking the penny on the first try. There are now two doors left once you have chosen, one of these is revealed to be a penny. Chances are you see the odds of the last door being a penny as well is the same odds as it being the million dollars.
There are many ways to explain why this is not the case. The one I found most useful is this.
Let's say the million dollars is behind door number one.
If you chose door number one you should stay. If you chose door number two you should switch. If you chose door number three you should switch.
Let's say the million dollars is behind door number two.
If you chose door number one you should switch. If you chose door number two you should stay. If you chose door number three you should switch.
Let's say the million dollars is behind door number three.
If you chose door number one you should switch. If you chose door number two you should switch. If you chose door number three you should stay.
Are you noticing a pattern? Most of the time you would be better off switching, in fact six out of nine times you would.
Where people get confused is when you pick your first door, you have a 66% chance of picking a penny, when the other penny is revealed, most believe the odds of the final door being the million dollars is only 33%. The confusing part is the fact you still chose a penny 66% of the time. This hasn't changed, so by removing a door you know for a fact is a penny, you have drastically increased your odds of choosing the money.
This challenge was on a game show called the Monty Hall show and resulted in this problem being referred to as The Monty Hall Problem. In the case of the game show, there was two goats behind the doors and a fancy car behind the last.
The problem was made famous when it was asked in Parade magazine to Marilyn vos Savant, a women with the Guinness World Record of the Highest IQ. Thousands of people responded back saying she was wrong. This resulted in many exploring it from mathematics and computer simulation, coming to the same conclusion. If you switch every time, you will win around 66% of the time.
There are many good YouTube videos on the problem, most of them will likely leave you more confused than when you started. I found this one to explain it best.
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