## Historical Remarks

Anyone who plays the same chance game over and over is really carrying out a simulation, and in this sense the process of simulation has been going on for centuries. As we have remarked, many of the early problems of probability might well have been suggested by gamblers' experiences. It is natural for anyone trying to understand probability theory to try simple experiments by tossing coins, rolling dice, and so forth. The naturalist Buffon tossed a coin 4040 times, resulting in 2048 heads and...

## Info

Then, since there is only one way to choose a set with no elements and only one way to choose a set with n elements, the remaining values of are determined by the following recurrence relation Theorem 3.4 For integers n and j, with 0 lt j lt n, the binomial coefficients satisfy Proof. We wish to choose a subset of j elements. Choose an element u of U. Assume first that we do not want u in the subset. Then we must choose the j elements from a set of n 1 elements this can be...

## Preface

Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continued to influence such early researchers as Huygens, Bernoulli, and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a well-established branch of mathematics that finds applications in every area of scholarly activity...

## Tree Diagrams

Example 1.10 Let us illustrate the properties of probabilities of events in terms of three tosses of a coin. When we have an experiment which takes place in stages such as this, we often find it convenient to represent the outcomes by a tree diagram as shown in Figure 1.8. A path through the tree corresponds to a possible outcome of the experiment. For the case of three tosses of a coin, we have eight paths wi, w2, , w8 and, assuming each outcome to be equally likely, we assign equal weight, 1...

## Random Numbers

We must first find a computer analog of rolling a die. This is done on the computer by means of a random number generator. Depending upon the particular software package, the computer can be asked for a real number between 0 and 1, or an integer in a given set of consecutive integers. In the first case, the real numbers are chosen in such a way that the probability that the number lies in any particular subinterval of this unit interval is equal to the length of the subinterval. In the second...

## P b

Bayes Probabilities Our original tree measure gave us the probabilities for drawing a ball of a given color, given the urn chosen. We have just calculated the inverse probability that a particular urn was chosen, given the color of the ball. Such an inverse probability is called a Bayes probability and may be obtained by a formula that we shall develop later. Bayes probabilities can also be obtained by simply constructing the tree measure for the two-stage...

## Exercises

1 Modify the program CoinTosses to toss a coin n times and print out after every 100 tosses the proportion of heads minus 1 2. Do these numbers appear to approach 0 as n increases Modify the program again to print out, every 100 times, both of the following quantities the proportion of heads minus 1 2, and the number of heads minus half the number of tosses. Do these numbers appear to approach 0 as n increases 2 Modify the program CoinTosses so that it tosses a coin n times and records whether...

## Card Shuffling

Much of this section is based upon an article by Brad Mann,28 which is an exposition of an article by David Bayer and Persi Diaconis.29 Given a deck of n cards, how many times must we shuffle it to make it random Of course, the answer depends upon the method of shuffling which is used and what we mean by random. We shall begin the study of this question by considering a standard model for the riffle shuffle. We begin with a deck of n cards, which we will assume are labelled in increasing order...

## Combinatorics

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 section and combinations in the next section. Before discussing permutations, it is useful to introduce a general counting technique that will enable us to solve a variety of counting problems, including the problem of counting the number of possible permutations of n objects. Consider an...

## Continuous Conditional Probability

In situations where the sample space is continuous we will follow the same procedure as in the previous section. Thus, for example, if X is a continuous random variable with density function f x , and if E is an event with positive probability, we define a conditional density function by the formula f x E 0, if x E. Then for any event F, we have The expression P F E is called the conditional probability of F given E. As in the previous section, it is easy to obtain an alternative expression for...

## There Has Been A Blizzard And Helen Is Trying To Drive From Woodstock To Tunbridge Which Are Connected Like The Top

We can calculate the numerator from our given information by P Hi n e P Hi P E Hi . 4.2 Since one and only one of the events hi, h2, , Hm can occur, we can write the probability of E as P e P Hi n e P h2 n e P Hm n E . Using Equation 4.2, the above expression can be seen to equal P hi p e hi P H P e h2 P Hm P E Hm . 4.3 Using 4.1 , 4.2 , and 4.3 yields Bayes' formula Although this is a very famous formula, we will rarely use it. If the number of hypotheses is small, a simple tree measure...

## D sin

Now we assume that when the needle drops, the pair 0, d is chosen at random from the rectangle 0 lt 0 lt n 2, 0 lt d lt 1 2. We observe whether the needle lies across the nearest line i.e., whether d lt 1 2 sin 0 . The probability of this event E is the fraction of the area of the rectangle which lies inside E see Figure 2.5 . Figure 2.5 Set E of pairs 9, d with d lt 1 sin 9. Now the area of the rectangle is n 4, while the area of E is The program BuffonsNeedle simulates this experiment. In...

## Charles Claims That He Can Distinguish Between Beer And

2 In how many ways can we choose five people from a group of ten to form a committee 3 How many seven-element subsets are there in a set of nine elements 4 Using the relation Equation 3.1 write a program to compute Pascal's triangle, putting the results in a matrix. Have your program print the triangle for n 10. 24A. W. F. Edwards, op. cit., p. ix. 25 J. Bernoulli, Ars Conjectandi Basil Thurnisiorum, 1713 . 5 Use the program BinomialProbabilities to find the probability that, in 100 tosses of a...

## Paradoxes

Much of this section is based on an article by Snell and Vanderbei.18 One must be very careful in dealing with problems involving conditional probability. The reader will recall that in the Monty Hall problem Example 4.6 , if the contestant chooses the door with the car behind it, then Monty has a choice of doors to open. We made an assumption that in this case, he will choose each door with probability 1 2. We then noted that if this assumption is changed, the answer to the original question...

## Suppose You Are Watching A Radioactive Source That Emits Particles At A Rate Described By The Exponential Density

We note that it is not the case that all continuous real-valued random variables possess density functions. However, in this book, we will only consider continuous random variables for which density functions exist. In terms of the density f x , if E is a subset of R, then The notation here assumes that E is a subset of R for which fE f x dx makes sense. Example 2.10 Example 2.7 continued In the spinner experiment, we choose for our set of outcomes the interval 0 lt x lt 1, and for our density...

## Explain Why It Is Not Possible To Define A Uniform Distribution Function On A Countably Infinite Sample Space

19 If A, B, and C are any three events, show that - P A n b - P B n C - P C n A P a n B n C . 20 Explain why it is not possible to define a uniform distribution function see Definition 1.3 on a countably infinite sample space. Hint Assume m w a for all w, where 0 lt a lt 1. Does m w have all the properties of a distribution function 21 In Example 1.13 find the probability that the coin turns up heads for the first time on the tenth, eleventh, or twelfth toss. 22 A die is rolled until the first...