A Needle on a Crack
 

A needle 2 inches long is dropped at random onto a floor made with wooden boards 2 inches in width, placed side by side. What is the probability that the needle falls across one of the cracks?

Calculus and very basic statistics required

Answer: [SPOILER] 2/pi[/SPOILER]

Ha! You're such a buffon!

Buffonade hasn't even started yet.

In Russian, this classical textbook question sounds even more contraversial than you can imagine:
"Какова вероятность что игла пересечет [URL="http://www.rustran.com/odr.php4?q=3&word=%EF%EE%EB%EE%E2%E0%FF+%F9%E5%EB%FC"][COLOR=#810081]половую щель[/COLOR][/URL]?"
Пол = 1) floor; 2) sex.
Pardon my Latin.

The given answer is only approximately correct.

He has forgotten that there is some probability that the needle will fall "point first" and stick in the floor like a flag pole.

[QUOTE=Wacky;194299]The given answer is only approximately correct.

He has forgotten that there is some probability that the needle will fall "point first" and stick in the floor like a flag pole.[/QUOTE]

Also, I wouldn't suppose there is much probability that the needle will land on the floor at all if this experiment is performed in outer space.

[QUOTE=Primeinator;194288]... falls across ...[/QUOTE][pedant]
"falls across" has not been properly defined. Does the needle have to land on the floor at all? Or just pass over (i.e. cross over, fall across the vertical line extending upwards from the cracks) it on the way down and land somewhere else?

What if the needle bounces and comes to rest outside the zone of the floor? Is the floor infinite and perfectly flat? Will the needle roll away because of a strong breeze? Is the geometry of the floor Euclidean? Is there a strong gravity field curving the floor so the needle can't rest on a flat surface? Are there magnets under the floor? Is the needle magnetic and is there a prevailing magnetic field (like on Earth)?

What if the needle falls into a crack? Is that considered "across"?

If the needle is 2 inches long and 1.9999 inches in diameter then how do you judge "across"? What if the needle is 1.9999 billion light years in diameter? Will the weight of the needle distort the floor?
[/pedant]

[QUOTE=Wacky;194299]He has forgotten that there is some probability that the needle will fall "point first" and stick in the floor like a flag pole.[/QUOTE]
And occasionally that will be into the crack. That is not across it.

I had not seen this puzzle until a few days ago and found it rather interesting. Although some peculiar caveats have been mentioned... I believe the problem assumes the needle will not plant itself like a flag and is asking the probability of it falling to the floor and then "settling" in a crack. Also, I would assume the diameter of the needle is approximated as being the same as that of the crack.

[QUOTE=Primeinator;194353]I had not seen this puzzle until a few days ago and found it rather interesting. [/QUOTE] And I first saw it a number of decades ago, but it remains a good problem to be solved. [QUOTE] Although some peculiar caveats have been mentioned...[/QUOTE] If you choose to "hang out" here, you should expect that. You are in the company of a number of pendantic individuals who relish the idea of finding any excuse to nullify "the obviously intended answer" [QUOTE] I believe the problem assumes the needle will not plant itself like a flag [/QUOTE] That was the author's intention [QUOTE] and is asking the probability of it falling to the floor and then "settling" in a crack. Also, I would assume the diameter of the needle is approximated as being the same as that of the crack.[/QUOTE] No, I think that you have it wrong. The ideal model is a needle that is a line segment of the exact length.

Given an infinite plane, ruled with uniformly spaced parallel lines, what is the probability that a randomly oriented diameter of a randomly placed circle will intersect one of the ruled lines if the diameter of the circle is the same as the spacing between the ruled lines.

[QUOTE=Wacky;194356]Given an infinite plane, ruled with uniformly spaced parallel lines, what is the probability that a randomly oriented diameter of a randomly placed circle will intersect one of the ruled lines if the diameter of the circle is the same as the spacing between the ruled lines.[/QUOTE]Yeah, what he said!

[quote=Wacky;194356]And I first saw it a number of decades ago, but it remains a good problem to be solved. [/quote]

It is a pretty way of expressing the average value of sin
between 0 and 90 or 180

David

(Pedants please note, I am talking in degrees Fahrenheit
down here):smile:

A lyric in this classic irritates me:
[URL]http://www.youtube.com/watch?v=ibINozjZ-Eg[/URL]

200 degrees that's why they call me "Mr Fahrenheit".
I'd be more impressed by "Mr Celsius".