The Original Problem
Many years ago, I posed a mathematical problem to a girlfriend. She could
not answer it, but nor could any of the other mathematicians I posed it to
over the years. So I thought I'd put it here on the web for others to have
a go at.
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?
Variants
At one time or another, in attempting to solve the original, the following
variants have also been discussed:
- A square pit
- An infinite pit
- X pebbles, where the aim is to have at least one pebble within a shape
formed by having the vertices at the other X-1 pebbles
- Y dimensions. (with the pit being the case for Y=2)
- Drop three pebbles into the pit. Then drop a 4th. What is the chance
that the 4th will fall inside a triangle formed by the first three?
- Place one pebble on the rim, in the 12 O'Clock position. Drop the other
three.
- Drop the pebbles, not so they have an equal chance of landing in any
area, but so their polar coordinates are random (equal chance of any
angle. equal chance of any radius)
- Drop circular pebbles of non-negligible size (eg each is 1/20th the
radius of the pit)
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.
Numerical Solutions
The answer to the original problem is 0.817 (to 3 sig fig).
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.)
Discussion and Feedback
Please add your solutions or other comments to
the Discussion Page
Last updated 2004年02月22日 by Douglas Reay