Problem 140 Solution

Consider that one end of the toothpick can fall anywhere between two lines with equal probability. Consider also that the angle formed by the line of the toothpick and the parellel lines is uniformally distributed. The probability can be set up as:

2/ $π$ * $∫$ (x=0 to 1) cos^-1(x) dx =

2/ $π$ * [x*cos^-1(x) - (1-x²)^1/2 for (x=0 to 1)] =

2/ $π$ * (0-0-0+1) =

2/ $π$ =~ 0.636620

Thanks to Extra Stuff: Gambling Rambling by Peter Griffin for this problem. See chapter 11.

Michael Shackleford, ASA, August 20 1999