It can be done with integrals or without integrals. If you're doing it with integrals, be careful with the limits. Remember that the maximum that both x and y can be is 1.
mugs
Lifer
- Apr 29, 2003
- 48,920
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I bet he wrote a program that generated 100,000 pairs of random numbers, divided, rounded, modded, and counted the evens and odds. (I assumed 100,000 pairs because of his level of precision)
I just did the same and got similar results. Now my question is - why?
Originally posted by: chuckywang
Original post now edited with hint. Enjoy.
You need to draw lines of the following functions:
y = 2x
y = 2/3x
y = 2/5x
y = 2/(2n + 1)x
and shade the arear between each alternative line. The area of triangle equals 1/2 * height * base. The heights of all these triangles are 1. To find base we need to plug x=1 into each equation and solve it for y, then find the difference between each y. The first equation is an exception since the line will intersect y plane instead of x plane. So you need to find the sum of the following series as n goes from 1 to infinity:
1 / (4n - 1) - 1 / (4n + 1).
Simplifying this a little we get:
8n / (16n^2 -1) with n=1,2,3...
I don't remember the formula for finding sums of inifinite series, but I believe there is one. Once you find the sum, you'll need to add 0.25 to it, since that's the first element in the series. The result should be your answer.
Originally posted by: Argo
Originally posted by: chuckywang
Original post now edited with hint. Enjoy.
You need to draw lines of the following functions:
y = 2x
y = 2/3x
y = 2/5x
y = 2/(2n + 1)x
and shade the arear between each alternative line. The area of triangle equals 1/2 * height * base. The heights of all these triangles are 1. To find base we need to plug x=1 into each equation and solve it for y, then find the difference between each y. The first equation is an exception since the line will intersect y plane instead of x plane. So you need to find the sum of the following series as n goes from 1 to infinity:
1 / (4n - 1) - 1 / (4n + 1).
Simplifying this a little we get:
8n / (16n^2 -1) with n=1,2,3...
I don't remember the formula for finding sums of inifinite series, but I believe there is one. Once you find the sum, you'll need to add 0.25 to it, since that's the first element in the series. The result should be your answer.
You are correct. It is a known summation that:
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ....
Your summation is 1/3 - 1/5 + 1/7 - 1/9 + 1/11 - 1/13 + ....
which is therefore equal to 1-pi/4.
Therefore, the final probability is 5/4-pi/4.
Originally posted by: FoBoT
Originally posted by: chuckywang
Originally posted by: FoBoT
that isn't a number
How so?
you know, a number? like 3 or 6 or 2324 ? do you mean 544 ?
or is that some fancy "new" math thing?
Wow FoBot, you're pretty retarded.
5/4 - pi/4 is the exact format
if you want a estimate, then the number is 1.25-(3.14/4) or ~.465ish.
No. It had to be exact. Argo won.
Fobot, you need to learn the difference between a "number" and an "integer"
Actually they were 10 million (if I remember correctly what I did last night), but I was too lazy to copy more digits.
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