Suppose there is a rectangular bench. Parallel lines are drawn on it with distance x between the parallel lines. If a needle of length x is dropped on the table what is the probability that it will intersect a parallel line. Plz show the solution as well
2006-08-02 03:49:02 · 6 answers · asked by aijazsahaf 1 in Science & Mathematics ➔ Mathematics
Assume that the center of the needle falls somewhere between the parallel lines (or assume there are multiple parallel lines; same thing), and that the location where the center falls is uniformly distributed between the lines. Assune further that the orientation of the fallen needle (its angle relative to a third parallel line passing through its center) is randomly distributed from zero to ninety degrees. (You have lots of symmetry here, so zero to ninety is okay.) Also assume that the two parallel lines are horizontal.
Let the center of the fallen needle be at position O, and let a be the distance from O to the nearest parallel line. The average value of a will be x/4.
If b is the acute angle formed by the needle and a parallel (horizontal) line passing through its center. If (x/2) sin b < a for any repetition of this experiment, then the fallen needle will intersect.
Since b is uniformly distributed between zero and ninety degrees, its average value is 45. On average, then, we have (x/2) sin 45 < x/4 ==> 2 sin 45 < 0 ==> sqrt(2) < 0, which is false. On average, the needle does not intersect.
To get this right, use this structure, let theta (your angle) range from 0 to 90, and integrate this thing from 0 to x/2. That will give you your probability.
2006-08-02 04:39:37 · answer #1 · answered by bpiguy 7 · 0⤊ 0⤋
The multiple parallel lines (assuming multiple lines) will be touched by the needle only when the angle of the needle creates a length perpendicular to the lines. At o degrees we assume it will not touch, ever. At 90 degrees it can not miss. If the angle is random, the average angle will be 45 degrees with a 1 / sq root of 2 length. The ratio should be related to the "touching time" (for the first .707 relative to the "non touching time" or .293. .707/.293 or 2.41or about 41.5% of the time for not touching. I only approx. the math, you'll want to do it more accurately.
2006-08-02 11:00:36 · answer #2 · answered by Steven A 3 · 0⤊ 0⤋
0%
The needle was dropped on the table....not the bench on which the lines were drawn. There are no lines on top of the table.
Good question, though.
2006-08-02 11:04:46 · answer #3 · answered by cakeeater0119 5 · 0⤊ 0⤋
Pi
2006-08-02 11:23:35 · answer #4 · answered by bob h 3 · 0⤊ 0⤋
none
2006-08-02 11:11:40 · answer #5 · answered by vebyllucs 3 · 0⤊ 0⤋
?
2006-08-02 12:09:21 · answer #6 · answered by Earl Morton 2 · 0⤊ 0⤋