TITLE: What is the chance that a rabbit won't fall off a table if you put it somewhere and it moves.
QUESTION [45 upvotes]: If you would put a rabbit randomly on a circular table with radius $r= 1$ meter and it moves $1$ meter in a random direction, what is the chance it won't fall off?
I tried to do this using integrals, but then I noticed you need a double integral or something and since I'm in the 5th form I don't know how that works.

REPLY [2 votes]: Let Roger Rabbit's location on the circular table be defined by the point $(X,Y)$, where random variables $X$ and $Y$ have a bivariate Uniform distribution inside the unit circle, with joint pdf $f(x,y)$:

By symmetry, we can assume, without loss of generality, that the rabbit jumps say 1m to the east (as Steve Jessop has noted above). Then, the probability that Roger Rabbit still lands on the table is $P[(X+1)^2 + Y^2 \leq 1]$:

All done.
Notes 

The Prob function used above is from the mathStatica package for Mathematica. As disclosure, I should add that I am one of the authors.