Hey

Seeing as my writing has dropped to a high school level, I figured it was time to start a blog. That's what all of the experts suggest, right? I've been taking almost strictly math and science courses while at Brown, and right now it's really hard for me to convert thoughts into words. For the time being I will post here about whatever I'm working on or thinking about, without any particular rhyme or reason. Right now my particular interests are competitive sailing, mathematics, coding, weight lifting, and maybe a little bit of meditation. So to start, two things:

1) As soon as I created the blog I had two views, and one was from Alaska. How cool is that!

2) I was discussing what I think is my favorite probability question with a friend a few days ago:

A teacher is returning graded tests to her class of n students, none of which are labelled with a student's name. If she returns the tests at random, what is the expected number of students who get the correct test?



Here expectation refers to the expected value, which is the sum of every value that the random variable X can take on, multiplied by the probability of obtaining that value.

For instance: flip a fair coin twice, what is the expected value if the value of heads=1 and tails=0?
E(X)=(0)(1/4)+(1)(1/2)+(2)(1/4)=1

I'm a little rusty on my formal probability, so I will just do this problem for a couple of cases, instead of proving it for the nth case.

Say that n=2:
then the E(X)=(0)(1/2)+2(1/2)=1
The probabilities here are determined as such because they only depend on the first person. If he gets the right test (P= 1/2) then the other person also gets the right test.

for n=3:
E(X)=(0)(1/3)+(1)(1/2)+(3)(1/6)=1
The trick here is that it's impossible to get 2 tests right and 1 wrong.

n=4:
E(X)=(1)(1/3)+(2)(1/4)+(4)(1/24)=1
(Using nCr/n! for probability of 2)


It's easy to see that this generalizes to the nth case (which is proved by linearity of expectation if I recall right). So no matter how many students the teacher has, she is expected to return just one test right!

Governing dynamics.