|How sweet it is!
||[Oct. 14th, 2004|11:12 am]
Warning! DO NOT READ IF MATHEMATICS CAUSES YOUR HEAD TO EXPLODE!!! The math isn't that bad, tough
We had a guest lecturer for class today for Probablility. His name is Dr. Ghahramani and he's the Dean of Arts and Sciences at John Hopkins. He wrote a new text book for the subject and he lectured us (along with a half of the Math/CS faculty that sat in on the lecture) on a couple of problems that have counter intuitive solutions, such as the birthday problem (in a class of 23 students, the probability of both having the same birthday is 0.507) and the "75% of all Triangles are obtuse" problem. The last problem we looked at was the Jailer's Paradox, which reads as follows.
"The jailer of a prison in which Alex, Ben, and Tim are held is the person, other than the judge, who knows which of these three prisoners is condemned to death, and which two will be freed. The prisoners know that exactly two of them will go free; they do not know which two. Alex has written a letter to is fianceé. Just in case he is not one of the two who will be freed, he wants to give the letter to a prisoner who goes free to deliver. So Alex asks the jailer to tell him the name of one of the two prisoners who will go free. The jailer refuses to give that information to Alex. To begin with, he is not allowed to tell Alex whether Alex goes free or not. Putting Alex aside, he argues that, if he reveals the name of a prisoner who will go free, then the probability of Alex dying increases from 1/3 to 1/2. He does not want to do that."
-Dr. Ghahramani, Fundamentals of Probability,, p41, 2004
Trust me, this story is going somewhere.
This problem has been debated and argued over since 1935. The most recent big hullabaloo happened in 1990 when they sent this problem to the person with the highest I.Q. recorded (I've forgotten her name). She wrote back saying that of course the probability of Alex dying stays at 1/3, regardless. She soon recieved over 2000 angry letters, 65% of which happened to come from Mathematicians.
We figured out that even if he did tell Alex the name of a prisoner that goes free, the probability of Alex dying is the same (if you really want the math behind this I can tell you about it.)
We concluded that Alex is a smart guy, at least smarter than the Jailer. However, if he gave his letter to Ben or Tim, not knowing which or both is going free, can Alex still get his message to his wife? Of COURSE HE CAN. My solution to the paradox is to give the letter to Ben. If Ben dies, then the letter to Alex's fianceé dies with him, which is fine since Alex would be alive to see his fianceé anyway! If Ben lives and Tim dies, then Alex can retrieve the letter from Ben after they get out of jail or just talk to his fianceé anyway. Last case, if Alex dies, Ben lives and gives the letter to Alex's fianceé. The same reasoning can be used to give Tim the letter.
Evidently, aside from myself, no one in the mathematics community had thought of this solution.
I raised my hand and asked the question "why doesn't he just give the letter to Ben?" Dr. Ghahramani looked at me with a tilted head, as if to say "Where have you been for the last half hour?!" He then said "Well what if Ben dies?" I replied with a sly smile "Then Alex is free to see his fianceé and the letter doesn't matter." He paused, and I heard people saying "Ohhhh" in a comprehensionary tone. Dr. Ghahramani looked down, trying to wrap is head around what I had said.
My classmates and the visiting professors (including Dr. Ghahramani) were duly impressed. I felt on top of the world. This was my shining moment. I was in my element and I rocked the mathematical Casbah. I was blushing I hope he mentions in his next edition of his textbook the solution to this problem. Remember, this is the frickin' DEAN of Arts and Sciences at John Hopkins. Well, at the end of class I had my picture taken with Dr. Ghaharamani, along with a couple of other blokes, and left for lunch. Dr. Moskol offered bonus points for asking good questions, and suffice it to say, I'm entitled to more than my fair share.
Eileen, if you could have only seen me, it was glorious.