If I drop four pebbles into a circular pit, what it the probability that one pebble will land inside a triangle formed from the other three pebbles? [The pebbles are of negligible size, the pit is flat, and the drop is done in such a way that each pebble has an equal chance of landing anywhere in the pit.] |
A numerical answer is trivial, but can you provide an analytic solution?
3 and 4 are generalisations. 1, 2, 7 and 8 are, I believe, different problems. 5 and 6 are more interesting. My hypothesis was that 5 would turn out to be different from the original, but that 6 is the same problem in a simpler form. I set out to provide support for this view by solving them numerically.
The answer to variant 1 is 0.824 (to 3 sig fig).
The answer to variant 5 is 0.323 (to 3 sig fig). (Which, it should be noted, is NOT precisely 1/4 of the original either.)
The answer to variant 6 is 0.859 (to 3 sig fig), which unfortunately blows my hypothesis out of the water.
The java code used to produce these numerical results is available under the GPL if you want to experiment around, or have a go at variants 2,3,4,7 or 8. (Depending on your browser, if you just want to browse not download the code, you may find this version easier to read.)
Last updated 2004-02-22 by Douglas Reay