Math Is Fun Forum

Each year has an equal probability of being the record-holder.  There are N years.  So for each year the probability of it being the record-holder is 1/N.  Let's say you have N balls in a bag, they are numbered 1 to N.  You reach into the bag and pull out a ball at random and without looking at the number on the ball, you write a random number between 1 and 1500 on the opposite side of the ball.  You do this for all of the N balls.  Which numbered ball will have the highest number?  It is easy to see that each ball has an equal chance of having the highest number.

Thank you for your response Bobbym.
This is kind of a strange problem because we are dealing with something varying around an average that is unknown.
True, the setting of 50% is somewhat arbitrary, but the first year that there is a record we assume that the rainfall (or temperature)
record is just as likely to be above as below the average, since we don't know what the average is.

I'm comparing the following 4 years only to the first year.
There are 2 possible sequences for the first 2 years, with only one sequence having the 2nd year setting a new record.  So we have a probability of 1/2 for the 2nd year setting a new record.  Similar reasoning shows that the probability of the 3rd year setting a new record is 1/3 and so on...

Use the difference between the sums to head towards an answer.

I enjoyed looking at your answer Bobbym, but I don't understand how you got

from

I'm not certain if this answer is correct, but it's the same answer from Mathematica.

I used the idea from gurthbruins to solve this problem.
This is a very hard problem.  I had printed out the problem on a sheet of
paper and decided to try to solve it.  First, we draw a straight line connecting
points A and M.  The point where this line intersects XY we'll call point Z.
We'll make another point called G which is the mid-point of AB.  The point of
tangency for the big circle we'll call W.  The center of the hexagon is the origin.
Notice that

We need to know what the coordinates of Z are.  The x-coordinate is

so the coordinates of point Z are

So we must solve the linear equation

simultaneously with the equations

in order to get the
coordinates of X and Y. Many, many steps later one ends of with
the x and y coordinates of ponts X and Y as follows...

using

to get the distance of XY we finally end
up with after many more steps.. drum roll please.....

as distance of XY.
Then afterwards be sure to take some Advil for the headache and
a tissue to wipe the blood now oozing from your eyes.

I got the same answer by a different method. I first calculated the centroid of a semi-circle of radius r centered on the y-axis.
the x-coordinate of the centroid will be zero, so we just need to calculate the y-coordinate...

which comes out to be

now we need to divide by the area so...

so the centroid is located at

The volume is equal to the area multiplied by the distance traveled by the centroid. So we have..

This is a difficult problem, but I think I have come up with the answer (hopefully).
You have a smaller cylinder intersecting a larger cylinder from the side.
Imagine that you have a 48" diameter cylinder and also a 58" diameter cylinder.
Imagine that you have a 24" diameter cylinder intersecting both.
Now, in order to find the volume of removed material all you have to do
is subtract the volume of intersection with the 48" cylinder from the
volume of intersection with the 58" cylinder. This all seems simple and
clear enough but how does one go about finding this?
Well, if you imagine the 24" cylinder intersecting the 58" cylinder you can
imagine that there will be a normal cylinder with a 'cap' on top.
The difficulty lies in finding the volume of this 'cap'.
The volume can be imagined as a rectangle moving through the 'cap'.
It's a bit hard to visualize but we need both the width and height
of the rectangle.
For the 24" cylinder looking down from above we have the relation

In this case, the x will serve as one of the sides of the rectangle.
For the 58" cylinder we have the relation

The height of the rectangle will be the part of h which is part of the
'cap' above the cylinder which is above 26.40075756.
We need the formula for the rectangle in terms of y since we will
be integrating with respect to y, so we end up with

for the area of the rectangle.
The integral will be from 0 to 12 and will be only 1/4 of the volume.
So the volume is equal to

Can anybody here integrate this?
I don't know how to integrate this so I cheated and used a computer program.
Now this is only the 'cap' so we need to add this to the cylinder
which is

So the total volume is

Now we do the same thing with the smaller cylinder

Total volume is

So the volume of removed material is

This is close to Bobbym's estimate of

Ok.  I believe I have a proof by contradiction. 

Well,
     Here's my attempt at a proof...

I'm happy to be back on mathisfun!