problem 394

[gelatine1]

i believe there is no rule how big he cuts his pieces. so that means his remaining part can always be lower then F after the first cut. so i don't really get what's E(x) is supposed to do and how it can be a non-integer value.
so i believe that i am misunderstanding the question or that there's missing a part in the question.

[wrongrook]

I wonder if you are thinking about the problem like a game where Jeff is choosing where to make the cuts?

In this case you are correct, and Jeff can "win" the game on his first move by cutting off most of the cake.

However, this is not the intention of the problem. It is better to think of it as a simulation where Jeff chooses his cuts by generating a (uniformly distributed) random number that points somewhere along the remaining pie border. It can therefore take 1 or 2 or more goes to eat the pie depending on what random numbers are chosen. The average number of goes is the expected value that you need to find as the answer.

I hope this helps, but apologies if I have misunderstood your misunderstanding!

[jaap]

Also, as I said in this thread, expected value has a specific mathematical meaning that you need to know to be able to answer questions like this.

[Marcus_Andrews]

i believe there is no rule how big he cuts his pieces. so that means his remaining part can always be lower then F after the first cut. so i don't really get what's E(x) is supposed to do and how it can be a non-integer value.
so i believe that i am misunderstanding the question or that there's missing a part in the question.

Expected value is basically "the average you get if you conduct the experiment infinitely many times" or "the average across all possible outcomes weighted by their probabilities."

For example, the expected value of a 6-sided die is 3.5. If you were to roll a single die, repeatedly, all day long, while taking the average, you'd get a number pretty close to 3.5. However, in this case you can calculate the EV directly by noting that (1/6)*1 + (1/6)*2 + (1/6)*3 + (1/6)*4 + (1/6)*5 + (1/6)*6 = 3.5.

3.5 is a decimal (i.e. it's not a whole integer value that you can roll on a die) but it is still technically the value of the expectation.

Hopefully this clarifies things a bit more. Keep in mind, though, that in the case of Problem 394, there are infinitely many ways to make the cuts.

[ffff0]

Can someone suggest reading materials for this problem? I hope that I simply don't know something obvious regarding continuous random variables.

[Marcus_Andrews]

Can someone suggest reading materials for this problem? I hope that I simply don't know something obvious regarding continuous random variables.

I hope this is not considered too much of a spoiler, but I would advise looking into probability density functions.

[kruzulis]

Could somebody please confirm that problem described below is equal to #394
Given n=1 and F=1/x repeat following procedure:
(
slice1 gets random value from range (0..n)
slice2 gets random value from range (0..n)
n gets new value: n=n-max(slice1,slice2)
if n is less that F then stop the procedure otherwise start with slicing again
)

Should find the expected value of number of times procedure is called before it stops.

[jaap]

Could somebody please confirm that problem described below is equal to #394

Yes, that seems correct.