3.1. PERMUTATIONS
91
(d) Set the expression found in part (c) equal to - log(2), and solve for d as a function of n, thereby showing that d 2(log 2) n .
Hint : If all three summands in the expression found in part (b) are used, one obtains a cubic equatio
3.1. PERMUTATIONS
89
7 Five people get on an elevator that stops at five floors. Assuming that each has an equal probability of going to any one floor, find the probability that they all get off at different floors. 8 A finite set has n elements. Show tha
3.1. PERMUTATIONS
87
king," just as in frustration solitaire. If the dealer goes through the 13 cards without a match he pays the players an amount equal to their stake, and the deal passes to someone else. If there is a match the dealer collects the play
3.1. PERMUTATIONS Houses Cats Mice Wheat Hekat 7 49 343 2401 16807 19607
85
The following interpretation has been suggested: there are seven houses, each with seven cats; each cat kills seven mice; each mouse would have eaten seven heads of wheat, each of
3.1. PERMUTATIONS Number of people 10 20 30 40 50 60 70 80 90 100 Probability that all birthdays are different .8830518 .5885616 .2936838 .1087682 .0296264 .0058773 .0008404 .0000857 .0000062 .0000003 Table 3.2: Birthday problem. We now turn to the topic
Chapter 3
Combinatorics
3.1 Permutations
Many problems in probability theory require that we count the number of ways that a particular event can occur. For this, we study the topics of permutations and combinations. We consider permutations in this secti
2.2. CONTINUOUS DENSITY FUNCTIONS
73
11 For examples such as those in Exercises 9 and 10, 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
2.2. CONTINUOUS DENSITY FUNCTIONS
69
Assignment of Probabilities
A fundamental question in practice is: How shall we choose the probability density function in describing any given experiment? The answer depends to a great extent on the amount and kind of
2.2. CONTINUOUS DENSITY FUNCTIONS
67
1-z
1-z
E E
1-z
1-z
Figure 2.19: Calculation of FZ .
0.03 0.025 0.02 0.015 0.01 0.005 20 40 60 80 100 120 f (t) = (1/30) e
- (1/30) t
Figure 2.20: Exponential density with = 1/30.
68
CHAPTER 2. CONTINUOUS PROBABILITY D
2.2. CONTINUOUS DENSITY FUNCTIONS
2
65
F (z) Z
1 0.8 0.6 0.4
1.75 1.5 1.25 1 0.75 0.5
f (z) Z
0.2
0.25
-1
-0.5
0.5
1
1.5
2
-1
-0.5
0
0.5
1
1.5
2
Figure 2.17: Distribution and density for Z = X 2 + Y 2 . E be the event cfw_Z z. Then the distribution funct
2.2. CONTINUOUS DENSITY FUNCTIONS
1
63
0.8
0.6
0.4
E.8
0.2
0.2
0.4
0.6
0.8
1
Figure 2.14: Calculation of distribution function for Example 2.14. When referring to a continuous random variable X (say with a uniform density function), it is customary to say
2.2. CONTINUOUS DENSITY FUNCTIONS
61
A glance at the graph of a density function tells us immediately which events of an experiment are more likely. Roughly speaking, we can say that where the density is large the events are more likely, and where it is s
2.2. CONTINUOUS DENSITY FUNCTIONS
59
Density Functions of Continuous Random Variables
Definition 2.1 Let X be a continuous real-valued random variable. A density function for X is a real-valued function f which satisfies
b
P (a X b) = for all a, b R.
f (x
2.2. CONTINUOUS DENSITY FUNCTIONS
57
game there may well be!), it is natural to assume that the coordinates are chosen at random. (When doing this with a computer, each coordinate is chosen uniformly from the interval [-1, 1]. If the resulting point does
2.2. CONTINUOUS DENSITY FUNCTIONS
55
This describes an experiment of dropping a long straw at random on a table on which a circle is drawn. Write a program to simulate this experiment 10000 times and estimate the probability that the length of the chord i
2.1. SIMULATION OF CONTINUOUS PROBABILITIES
53
7 For Buffon's needle problem, Laplace9 considered a grid with horizontal and vertical lines one unit apart. He showed that the probability that a needle of length L 1 crosses at least one line is p= 4L - L2
2.1. SIMULATION OF CONTINUOUS PROBABILITIES
49
1. To simulate this case, we choose values for x and y from [-1, 1] at random. Then we check whether x2 + y 2 1. If not, the point M = (x, y) lies outside the circle and cannot be the midpoint of any chord, a
2.1. SIMULATION OF CONTINUOUS PROBABILITIES
1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2
47
Figure 2.7: Sum of two random numbers. [a, b] approximates the probability that a X b. But the sum of the areas of these bars also approximates the integral
b
f (x) dx .
a
Th
2.1. SIMULATION OF CONTINUOUS PROBABILITIES
45
1/2
d
Figure 2.4: Buffon's experiment.
d
1/2
E
0 0 /2
Figure 2.5: Set E of pairs (, d) with d <
1 2
sin .
Now the area of the rectangle is /4, while the area of E is
/2
Area =
0
1 1 sin d = . 2 2 1/2 2 = . /
2.1. SIMULATION OF CONTINUOUS PROBABILITIES
y
43
y = x2 1
E
x 1
Figure 2.2: Area under y = x2 . for this simple region we can find the exact area by calculus. In fact,
1
Area of E =
0
x2 dx =
1 . 3
We have remarked in Chapter 1 that, when we simulate an e
Chapter 2
Continuous Probability Densities
2.1 Simulation of Continuous Probabilities
In this section we shall show how we can use computer simulations for experiments that have a whole continuum of possible outcomes.
Probabilities
Example 2.1 We begin by
1.2. DISCRETE PROBABILITY DISTRIBUTIONS
39
26 Two cards are drawn successively from a deck of 52 cards. Find the probability that the second card is higher in rank than the first card. Hint : Show that 1 = P (higher) + P (lower) + P (same) and use the fac
1.2. DISCRETE PROBABILITY DISTRIBUTIONS
37
less than 85 percent lost one leg. What is the minimal possible percentage of those who simultaneously lost one ear, one eye, one hand, and one leg?22 *17 Assume that the probability of a "success" on a single ex
1.2. DISCRETE PROBABILITY DISTRIBUTIONS
35
We see that these two mathematicians arrived at two very different ways to solve the problem of points. Pascal's method was to develop an algorithm and use it to calculate the fair division. This method is easy t
1.2. DISCRETE PROBABILITY DISTRIBUTIONS
33
3
0
0
0
2 Number of games B has won 1
8
16
32
64
20
32
48
64
0
32
44
56
64
0
1 2 Number of games A has won
3
Figure 1.9: Pascal's table. quicker and neater, which I would like to tell you here in a few words: for
1.2. DISCRETE PROBABILITY DISTRIBUTIONS
31
be 0, 1, or 2, and hence we should assign equal probabilities for these three possible outcomes.16 Cardano chose the correct sample space for his dice problems and calculated the correct probabilities for a varie