For examples such as those in Exercises Exercise and Exercise, it might seem that at least you should not have to wait on average more than 10 minutes if the average time between occurrences is 10 minutes. Alas, even this is not true. To see why, consider the following assumption about the times between occurrences. Assume that the time between occurrences is 3 minutes with probability .9 and 73 minutes with probability .1. Show by simulation that the average time between occurrences is 10 minutes, but that if you come upon this system at time 100, your average waiting time is more than 10 minutes.
Take a stick of unit length and break it into three pieces, choosing the break points at random. (The break points are assumed to be chosen simultaneously.) What is the probability that the three pieces can be used to form a triangle?
Hint: The sum of the lengths of any two pieces must exceed the length of the third, so each piece must have length [math] \lt 1/2[/math]. Now use Exercise(g).
Take a stick of unit length and break it into two pieces, choosing the break point at random. Now break the longer of the two pieces at a random point. What is the probability that the three pieces can be used to form a triangle?
Choose independently two numbers [math]B[/math] and [math]C[/math] at random from the interval [math][-1,1][/math] with uniform distribution, and consider the quadratic equation
Find the probability that the roots of this equation
- are both real.
- are both positive.
Hints: (a) requires [math]0 \leq B^2 - 4C[/math], (b) requires [math]0 \leq B^2 - 4C[/math], [math]B \leq 0[/math], [math]0 \leq C[/math].
At the Tunbridge World's Fair, a coin toss game works as follows.
Quarters are tossed onto a checkerboard. The management keeps all the quarters, but for each quarter landing entirely within one square of the checkerboard the management pays a dollar. Assume that the edge of each square is twice the diameter of a quarter, and that the outcomes are described by coordinates chosen at random. Is this a fair game?
Three points are chosen at random on a circle of unit circumference. What is the probability that the triangle defined by these points as vertices has three acute angles? Hint: One of the angles is obtuse if and only if all three points lie in the same semicircle. Take the circumference as the interval [math][0,1][/math]. Take one point at 0 and the others at [math]B[/math] and [math]C[/math].
Write a program to choose a random number [math]X[/math] in the interval [math][2,10][/math] 1000 times and record what fraction of the outcomes satisfy [math]X \gt 5[/math], what fraction satisfy [math]5 \lt X \lt 7[/math], and what fraction satisfy [math]x^2 - 12x + 35 \gt 0[/math]. How do these results compare with Exercise?
Write a program to choose a point [math](X,Y)[/math] at random in a square of side 20 inches, doing this 10,00 times, and recording what fraction of the outcomes fall within 19 inches of the center; of these, what fraction fall between 8 and 10 inches of the center; and, of these, what fraction fall within the first quadrant of the square. How do these results compare with those of Exercise?