Wednesday evening we drove 550 miles from Missoula to Portland to spend the holiday with Hilary’s friend Kathleen. We crammed a lot into a few days and Kathleen was gracious enough to be our local tour guide. These pictures are from Thanksgiving Day.
You are a contestant on the game show “Lets Make a Deal” and host Monty Hall presents you with a scenario:
Three closed doors, with a shiny new car randomly placed behind only one.
Monty instructs you to pick one door you think has the car behind it. You pick a door, suppose it is the first door, door #1.
Monty: “Now, before we open door #1, lets see what’s behind door #3!”…..no car.
(Monty knows where the car is and would never open that door)
At this point in the game, doors #1 and #2 remain unopened and Monty generously gives you the option of switching from your original choice to door #2.
Since you really want that car, some questions run through your head.
“Should I switch doors or just stay with my original choice? Are my odds of winning even affected by whether I switch or not?”
With two doors unopened doors left, it is intuitive to think that the probability the car is behind either door left is one-half, and therefore it doesn’t matter if you stay or switch doors. Might as well flip a coin to determine which door you end up with.
The preceding thought process is common. Unfortunately, it is also completely wrong and might cost you a car. The truth is you are twice as likely to win if you switch doors.
When Monty revealed a door guaranteed to not have the car, he provided you some valuable information.
To see why this is, start the game over.
1 out of every 3 times you play the game you will pick the winning door to start with. If this happens and you switch doors, you will end up losing.
2 out of every 3 times you will pick a losing door with your initial pick. Here, switching will always end up with you winning the car. This fact is a little more subtle. Monty guarantees you will win if you switch when he revealed what was the only remaining losing door. The only door left to switch to is the one with the car.
In the long run, switching every single time you play will allow you to win twice as often as if you never switched. Simply because you are twice as likely to pick a losing door initially as you are to pick a winning door.
The Monty Hall Problem is an example of the difference conditional probability makes. Given that Monty opens what he knows to be a losing door, a strategy where you always switch is the better one.
For more insight on this problem, check out Wikipedia.
Pictures of a rad Saturday in Missoula playing with frisbees.
The the first picture links to a gallery. The last three are pictures that were too big to include.
