Advanced book on Mathematics Olympiad

786 Combinatorics and Probability

933.The pair(p, q)is chosen randomly from the three-dimensional domainC×intC,
which has a total volume of 2π^2 (here intCdenotes the interior ofC). For a fixedp, the
locus of pointsqfor whichRdoes not have points outside ofCis the rectangle whose
diagonal is the diameter throughpand whose sides are parallel to the coordinate axes
(Figure 112). If the coordinates ofpare(cosθ,sinθ), then the area of the rectangle is
2 |sin 2θ|.

x

y

p

θ

Figure 112

The volume of the favorable region is therefore

V=

∫ 2 π

0

2 |sin 2θ|dθ= 4

∫π/ 2

0

2 sin 2θdθ= 8.

Hence the probability is

P=

8

2 π^2

=

4

π^2

≈ 0. 405.

(46th W.L. Putnam Mathematical Competition, 1985)

934.Mark an endpoint of the needle. Translations parallel to the given (horizontal) lines
can be ignored; thus we can assume that the marked endpoint of the needle always falls
on the same vertical. Its position is determined by the variables(x, θ ), wherexis the
distance to the line right above andθthe angle made with the horizontal (Figure 113).
The pair(x, θ )is randomly chosen from the region[ 0 , 2 )×[ 0 , 2 π). The area of this
region is 4π. The probability that the needle will cross the upper horizontal line is

1

4 π

∫π

0

∫sinθ

0

dxdθ=

∫π

0

sinθ

4 π

dθ=

1

2 π

,

which is also equal to the probability that the needle will cross the lower horizontal line.
The probability for the needle to cross either the upper or the lower horizontal line is
thereforeπ^1. This gives an experimental way of approximatingπ.
(G.-L. Leclerc, Comte de Buffon)