When you teach at 8am the week after daylight savings time you get to see remnants of the sunrise. Combined with the light snow overnight made for a pleasant bike ride to campus.
The Monty Hall Problem is probably the most famous example of how the math of uncertainty can be unintuitive and rather surprising.
Time to use a bit of imagination.
You are a contestant on the game show “Lets Make a Deal” and the host, Monty Hall, presents you with the following scenario:
There are three closed doors up on the stage. One door is hiding shiny new car. All three doors look exactly the same and you have no idea where the car is.
You want that car. You NEED that car.
Monty instructs you to pick one door. After much deliberation you decide door #1.
The game continues:
Monty: “Now, before we open door #1, lets see what’s behind door #3!”
No car. A goat..Interesting.
Now there are two remaining unopened doors, #1 and #2. You know one of them is hiding a car.
The game continues:
Monty: “Do you want to stick with door #1 or switch to door #2?”
You think: “Should I switch doors? Should I stay? With 2 doors left does it even matter what I do?”
It is common to think that the probability the car is behind either of the remaining doors is one-half you have an equally likely chance of winning no matter what.
The surprising truth is that the preceding thought process is completely wrong. The reality is, loosely speaking, you are twice as likely to win if you switch doors. WTF?
Before you decide to stay or switch, Monty always reveals a door that does not have the car. This is extremely valuable information!
To see why, think about the beginning of the game and your first choice of doors.
Initially, you have a one-third chance of picking the winning door. Under this scenario, if you switch doors, you will end up with a losing door. This is relatively obvious.
Similarly, your chance of picking the wrong door initially is two-thirds. Here, if you switch doors, you will always end up with you winning the car. This fact is subtle and key to understanding this problem.
Remember how we said Monty revealing a losing door was important? If you started with the wrong door, and Monty revealed the other wrong door, then only unopened door left is the one with the car!
What this means is if your original pick was wrong and you then switch to the remaining unopened door, you are guaranteed to win.
It might be helpful to draw a crude picture of 3 doors and see what I am talking about visually.
Since, you are twice as likely to pick a losing door initially, you are twice as likely to win if your strategy is to always switch doors.
Does switching doors guarantee you will win? Unfortunately, no. For an individual game the outcome is always uncertain. In the middle of the game you never know if your initial pick was right or not and there is no way to change this. (bribery?)
The mathematical interpretation of you are “twice as likely to win” means that if you played this game many many many times and switched doors every time, you would win about twice as many games as you would if you played those games and never switched.
The challenging nature of this problem lies in the often unintuitive realm known as conditional probability.
The question is not “What is the probability that I win if I switch doors?”
The exact question is “What is the probability that I will win given that Monty helped me out tremendously by revealing a losing door?”.
Two-thirds! Always switch!
(Don’t get mad at math if you switch doors and end up losing. For that you can thank randomness and uncertainty.)
For more insight on this problem, check out Wikipedia.
In the last week I have
1) Finished school (M.S. in Statistics from Montana State University)
2) Biked from Bozeman to Missoula (330 miles)
3) Packed my stuff up and moved to Missoula
It has been a big week, and with the Sasquatch Music Festival this weekend the action continues.
Thanks to everyone that helped me get through the last three years. It is crazy to me how fast time passes. Days go by like hours used to when I was little.
The bike trip was epic. I will put up more pictures later. Now I have some unpacking to do.
Last Friday we had the annual Christmas Party. A plethora of goodies always insures a gala event. It was the final hurrah before the cold, stark reality of finals week set in.
Fall goes by quickly in Bozeman. School has been keeping me well occupied these days, but I have managed to take a few pictures of the season on occasion.
I’ve been a little weather obsessed lately. To compliment the snowy pictures here’s what things looked liked around campus last Thursday evening. It was warm, I was in shorts and a sweatshirt. Being from North Carolina it is odd when it stays light until 9 pm but none of the trees have any leaves on them. The grass is getting green though, there is some color emerging from the whites and browns of winter.
It is supposed to be cold for the next couple of days but I know spring can’t be too far away.
I went to Yellowstone over the weekend and took a bunch of pictures but the interweb is being testy so those are going to have to wait.
I’ll be done with my first year of graduate school in about a week. Which is an exciting thought, although it means that I have a lot to do this week. Then about a 10 day break and I’ll teach Stat 216 again for the first summer session, and then almost 2 months of freedom! Hopefully during this time I’ll make it over to see Davy in Tokyo and back to Asheville in August.
It has been warm here for the past week, the grass is bright green and the trees are finally starting to show some color as well.
I hope everyone is doing well and I’ll try to put those pictures up ASAP.
I recently found out that I will I have a job for the first summer school session here teaching Statistics 216. It is the introductory statistics class and what I’m currently teaching. I really enjoy the material and feel like I’m learning a lot to get to the level where I feel comfortable lecturing on it.
It also pays fairly good money (especially for Bozeman). Hopefully it will be all the work I have to do in the summer. I’m also looking forward to having the opportunity to teach this material again so soon, I feel like it will be easier than if I waited a whole summer. Furthermore, the 1st summer sessions ends on June 29th and school doesn’t start again until August 27th! Now I can make definite plans for the summer.
p.s. We have about 7 inches on new snow on the ground as of right now and Bridger is reporting 15 new inches in the last 24 hours. Reminds me I’m living in Montana and it is not quite springtime yet.
School tomorrow. I can handle that I believe. I’m going to start attending a professor’s Stat 216 class so I can see how someone else teaches the material (which can be a little tricky) and learn a few things while I’m at it. Its at 8 am but I’ve been going to bed early enough that its not really a big deal.
Things are going well here. I went to ultimate today for the first time in Bozeman since October or something like that. Since before it got cold and they moved it into a gym, which wasn’t too exciting to me. But I figure ultimate in a gym is better than no ultimate at all. School, snowboarding, and climbing have been running my life since I got back into town.
As I mentioned to Davy I watched this movie “Born into Brothels”. Its a very interesting movie, with a lot of information about India. And it has a very happy ending, which is quite nice. I’m a fan of the happy ending. It also has a lot about photography in it, and it won an Oscar in 2004 for best documentary. Which hopefully An Inconvienent Truth will win this year if people have any sense at all (I’m still debating on this…the people having sense thing).
Life is very good right now. A continuous amount of snowfall between now and April and it will be ridiculous.
Come and visit me in Bozeman. The End
Today is the first day of classes at MSU.Â I have tuesdays and thursdays free so I don’t actually start until tomorrow. I have class at 9, I teach at 11, a class at 1:10, and then I’m done. Not a bad day really, I remember I did the mon, wed, fri schedule in undergrad and it made for some hellish days, but I think this schedule will be quite reasonable.
Its good that I have a free day today because I have a to-do list with approximately a million and one items on it. I’m not going to Bridger today so that I can make a dent in this list. Which will put some pressure on me to actually accomplish things because if I don’t, well, then I should have been at Bridger. (Which got 3″ of new snow yesterday which made for a lovely day).