Advanced book on Mathematics Olympiad

Combinatorics and Probability 783

The favorable cases consist of the region

Df=

{

(x, y)∈D|x+y≤ 1 ,xy≤

2

9

}

.

This is the set of points that lie below both the linef(x)= 1 −xand the hyperbola
g(x)= 92 x.

The required probability isP =AreaArea(D(D)f). The area ofDis 1. The area ofDfis
equal to

∫ 1

0

min(f (x), g(x))dx.

The line and the hyperbola intersect at the points(^13 ,^23 )and(^23 ,^13 ). Therefore,

Area(Df)=

∫ 1 / 3

0

( 1 −x)dx+

∫ 2 / 3

1 / 3

2

9 x

dx+

∫ 1

2 / 3

( 1 −x)dx=

1

3

+

2

9

ln 2.

We conclude thatP=^13 +^29 ln 2≈ 0 .487.
(C. Reischer, A. Sâmboan,Culegere de Probleme de Teoria Probabilita ̧ ̆tilor ̧si Statis-
tic ̆a Matematica ̆(Collection of Problems of Probability Theory and Mathematical Statis-
tics), Editura Didactica ̧ ̆si Pedagogica, Bucharest, 1972) ̆

928.The total region is a square of sideβ. The favorable region is the union of the
two triangular regions shown in Figure 109, and hence the probability of a favorable
outcome is

(β−α)^2

β^2

=

(

1 −

α

β

) 2

.

Figure 109

(22nd W.L. Putnam Mathematical Competition, 1961)

929.Denote byx, respectively,y, the fraction of the hour when the husband, respectively,
wife, arrive. The configuration space is the square