6.3 Probability 319Solution.The center of the coin falls on some tile. For the coin to lie entirely on that
tile, its center must fall inside the dotted square of side lengthl− 2 ·d 2 =l−dshown
in Figure 42. This happens with probability
P=(l−d)^2
l^2.
For the game to be fair,Pmust be equal to^12 , whence the relation thatdandlshould
satisfy is
d=1
2
( 2 −
√
2 )l. Example.What is the probability that three randomly chosen points on a circle form an
acute triangle?
Solution.The fact that the triangle is acute is equivalent to the fact that each of the arcs
determined by the vertices is less than a semicircle.
Because of the rotational symmetry of the figure, we can assume that one of the
points is fixed. Cut the circle at that point to create a segment. In this new framework,
the problem asks us to find the probability that two randomly chosen points on a segment
cut it in three parts, none of which is larger than half of the original segment.
Identify the segment with the interval[ 0 , 1 ], and let the coordinates of the two points
bexandy. Then the possible choices can be identified with points(x, y)randomly
distributed in the interior of the square[ 0 , 1 ]×[ 0 , 1 ]. The area of the total region is
therefore 1. The favorable region, namely, the set of points inside the square that yield
an acute triangle, is
{
(x, y)
∣∣
∣∣ 0 <x<^1
2,
1
2
<y<1
2
+x}
∪
{
(x, y)∣∣
∣∣^1
2
<x< 1 ,x−1
2
<y<1
2
}
.
The area of this region is^14. Hence the probability in question is^14.
As an outcome of the solution we find that when cutting a segment into three random
parts, the probability that the three segments can be the sides of an acute triangle is^14.
927.What is the probability that the sum of two randomly chosen numbers in the interval
[ 0 , 1 ]does not exceed 1 and their product does not exceed^29?
928.Letαandβbe given positive real numbers, withα<β. If two points are selected
at random from a straight line segment of lengthβ, what is the probability that the
distance between them is at leastα?
929.A husband and wife agree to meet at a street corner between 4 and 5 o’clock to go
shopping together. The one who arrives first will await the other for 15 minutes,
and then leave. What is the probability that the two meet within the given time
interval, assuming that they can arrive at any time with the same probability?