An infinately large grid is formed from lines drawn 1 inch apart horizonatally and vertically. A one inch pin is droped on this grid at a random location and a random angle.
What is the chance that the pin will cross a line?
2006-07-20 15:38:21 · 7 answers · asked by Michael M 6 in Science & Mathematics ➔ Mathematics
The probability is 3/pi (95.493%).
Prove: suppose the pin is dropped under an angle a with the horizontal. The pin then has "width" cos a and "height" sin a. (We assume that 0 <= a <= 90 degrees.)
The probability that is crosses a vertical line is equal to the width; the probability that it crosses a horizontal line is equal to the height. The probability that is crosses one of the lines is
[1] ... p(a) = sin a + cos a - sin a cos a.
In order to find the overall probability we must add the probabilities together for all angles a. We must consider a in the range from 0 to 90 degrees, or 0 to pi/2 radians.
P = {integral of p(a)da for a = 0..pi/2} / {pi/2}
This is equal to {3/2} / {pi/2} = 3/pi.
2006-07-20 16:10:44 · answer #1 · answered by dutch_prof 4 · 9⤊ 2⤋
2 to 1
2006-07-20 22:41:20 · answer #2 · answered by wizard 4 · 0⤊ 0⤋
seriously it is like 99.999999999999999...% likely that it will hit a line. If you were to conduct millions and millions of trials it is highly unlikely that the pin would ever fall on the grid without touching a line. Why? First of all, the center of the pin would need to fall directly in the center of the 1 by 1 inch square which is like in in a million. Not only would this need to happen, but even if the diameter of the pin is negligible it would have to fall almost directly on the diagonal of the 1 by 1 inch square. So while it is not impossible to land the pin without crossing a line, it is Whit out a doubt nearly impossible!
2006-07-20 23:05:53 · answer #3 · answered by scoobedue81 1 · 0⤊ 0⤋
depends on if randomly is selected by a computer or not. because if it is you can calculate exactly where it will land. but either way it is 100% chance of happening. because it will land and then fall over. once that happens, if the pin is longer than 1 inch, which it probibly is, it will cross the lines no matter what.
2006-07-20 23:07:07 · answer #4 · answered by Anonymous · 0⤊ 0⤋
50-50.
2006-07-21 01:37:26 · answer #5 · answered by san_pellegrino 4 · 0⤊ 0⤋
as random as this question is it may or may not be as simple as 50%. my guess is 98009% to1.0001% it will or it wont doesnt cut it in an infinite world but everything else may or may not.
2006-07-20 23:10:53 · answer #6 · answered by cerberus 2 · 0⤊ 0⤋
50% - it either will, or it won't.
2006-07-20 22:44:07 · answer #7 · answered by Stuart 7 · 0⤊ 0⤋