2 Interpretations of Probability
In addition to the many formal applications of probability theory, the concept of probability enters our everyday life and conversations.
- It probably will rain tomorrow.
- It is very likely that the bus will arrive late.
- The chances are good he will win the game.
Despite the fact that the concept of probability is such a common and natural part of our experience, no single scientific interpretation of the term probability is accepted by all statisticians.
Frequency interpretation: In many problems, the probability that some specific outcome of a process will be obtained can be interpreted to mean the relative frequency with which that outcome would be obtained if the process were repeated a large number of times under similar conditions.
Example: the probability of obtaining a head when a coin is tossed is approximately 1/2. Difficulties: Large number, similar conditions.
The frequency interpretation applies only to a problem in which there can be, at least in principle, a large number of similar repetitions of a certain process. Many important problems are not of this type. For example, the probability that a particular medical research project will lead to the development of a new treatment for a certain disease within a specified period of time.
Remark: The mathematical theory of probability can be usefully applied, regardless of which interpretation of probability is used in a particular problem. The theories and techniques that will be presented in this book have served as valuable guides and tools in almost all aspects of the design and analysis of effective experimentation.
2 Experiments and Events
The term “experiment” is used in probability theory to describe virtually every process whose outcome is not known in advance with certainty. Here are some examples.
(1) In an experiment in which a coin is to be tossed 10 times, the experimenter might want to determine the probability that at least four heads will be obtained.
(2) In an experiment in which the air temperature at a certain location is to be observed every day at noon for 90 successive days, a person might want to determine the probability that the average temperature during this period will be less than some specified value.
(3) In evaluating an industrial research and development project at a certain time, a person might want to determine the probability that the project will result in the successful development of a new product within a specified number of months.
The interesting feature of an experiment is that each of its possible outcomes can be specified before the experiment is performed, and probabilities can be assigned to various combinations of outcomes that are of interest.
3 Set Theory
3.1 The sample space
The collection of all possible outcomes of an experiment is called the sample space of the experiment. Each possible outcome is called a point, or an element, of the sample space. The sample space can be classified into two types: countable and uncountable. A sample space is countable if there is a one-to-one correspondence between the elements of A and the set of natural numbers {1, 2, 3, . . .}. Otherwise, it is uncountable.
3.2 Relations of set theory
Let S denote the sample space of some experiment. Then each possible outcome s of the experiment is said to be a member of the space S, or to belong to the space S. The statement that s is a member of S is denoted symbolically by the relation s 2 S.
An event C is a certain subset of possible outcomes in the space S. We say the event C occurs if the outcome of the experiment is in the set C.
Example 3.1 A six-sided die is rolled. The sample space is S = {1, 2, 3, 4, 5, 6}. Let A = {2, 4, 6} denote the event A that an even number is obtained, and B = {4, 5, 6}denote the event that a number greater than 3 is obtained. We say A occurs if we get the number 2 in a roll.
It is said that an event A is contained in another event B if every outcome that belongs to the subset defining the event A also belongs to the subset defining the event B. The relation is denoted by A ½ B. Mathematically, we have
A ½ B , x 2 A ) x 2 B (containment)
If two events are so related that A ½ B and B ½ A, then A =B, which means that A and B must contain exactly the same sample points. Mathematically, we have
A = B , A ½ B and B ½ A (equality)
A subset of S that contains no outcomes is called the empty set, or null set, and it is denoted by the symbol ;. For each event, it is true that ;½ A ½ S.
3.3 Operations of set theory
Given any two events (or sets) A and B, we have the following elementary set operations:
Union: The union of A and B, written A [ B, is the set of elements that belong to either A or B or both:
A [ B = {x : x 2 A or x 2 B}
Intersection: The intersection of A and B, written A \ B, is the set of elements that belong to both A and B:
A \ B = {x : x 2 A and x 2 B}
Complements: The complement of A, written Ac, is the set of all elements that are not inA:
Ac= {x : x A}
Disjoint events
It is said that two events A and B are disjoint, or mutually exclusive, if A and B have no outcomes in common. It follows that A and B are disjoint if and only if A\B=;.
The events A1, . . . ,Anor the events A1,A2, . . .are disjoint if for every i 6= j, we have that Aiand Ajare disjoint, that is, Ai\ Aj= ; for all i 6= j.
Example 3.2 (Event operations) Consider the experiment of selecting a card at random from a standard deck and noting its suit: clubs (C), diamond (D), hearts (H), or spades (S). The sample space is
S = {C,D,H, S},
and some possible events are
A = {C,D}, and B = {D,H, S}.
From these events we can form
A [ B= {C,D,H, S}, A \ B ={D}, and Ac= {H, S}.
Furthermore, notice that A [ B =S and (A [ B)c= ;.
4 The Definition of Probability
4.1 Axioms and Basic Theorems
In order to satisfy the mathematical definition of probability, P(A), the probability assigned to event A, must satisfy three specific axioms.
Axiom 1 For every event A, P(A) ¸ 0. In words, the probability of every event must be nonnegative.
Axiom 2 P(S) = 1. If an event is certain to occur, then the probability of that event is 1.
Axiom 3 For every infinite sequence of disjoint events A1,A2, . . .,
The mathematical definition of probability can now be given as follows: A probability distribution, or simply a probability, on a sample space S is a specification of numbers P(A) that satisfy Axioms 1, 2, and 3.
Some Theorems
Theorem 4.1 P(;) = 0.
Theorem 4.2 For every finite sequence of n disjoint events A1, . . . ,An,
Theorem 4.3 For every event A, P(Ac) = 1 −P(A).
Theorem 4.4 If A ½ B, then P(A) ≤P(B).
Theorem 4.5 For any event A, 0 ≤P(A) ≤1.
Theorem 4.6 For every two events A and B,
P(A [ B) = P(A) + P(B) −P(AB).
5 Finite sample spaces
5.1 Requirements of probabilities
In an experiment of which the sample space contains only a finite number of points s1, . . . , sn, a probability distribution on S is specified by assigning a probability pito each point si2 S. In order to satisfy the axioms for a probability distribution, the numbers p1, . . . , pnmust satisfy the following two conditions:
pi ¸ 0, for i =1, . . . , n
and
The probability of each even A can then be found by adding the probabilities piof all outcomes sithat belong to A.
Example 5.1 Consider an experiment in which five fibers having different lengths are subject to a testing process to learn which fiber will break first. Suppose that the lengths of the five fibers are 1,2,3,4, and 5 inches, respectively. Suppose also that the probability that any given fiber will be the first to break is proportional to the length of that fiber.
A sample space S containing n outcomes s1, . . . , snis called a simple sample space if the probability assigned to each of the outcomes s1, . . . , snis 1/n. If an event A in this simple sample space contains exactly m outcomes, then
5 Finite sample spaces
Example 5.2 (Tossing coins) Suppose that three fair coins are tossed simultaneously. We shall determine the probability of obtaining exactly two heads.
6 Counting Methods
In many experiments, the number of outcomes in S is so large that a complete listing of these outcomes is too expensive, too slow, or too likely to be incorrect to be useful. In such an experiment, it is convenient to have a method of determining the total number of outcomes in the space S and in various events in S without compiling a list of all these outcomes.
Multiplication rule
For example, suppose that there are three different routes from city Ato city B and five different routes from city B to city C. Then the number of different routes from A to C that pass through B is 3 × 5 = 15.
6.2 Permutations
Suppose that k cards are to be selected one at a time and removed from a deck of n cards (k = 1, 2, . . . , n). Then each possible outcome of this experiment will be a permutation of k cards from the deck, and the total number of these permutations will be Pn,k= n(n − 1) · · · (n − k + 1). This number Pn,kis called the number of permutations of n elements taken k at a time.
If k =n, Pn,n= n(n − 1) · · · 1 = n!. Note that 0! = 1.
Example 6.1 (Arranging books) Suppose that six different books are to be arranged on a shelf. The number of possible permutations of the books is 6! = 720.
Example 6.2 (Ordered sampling without replacement) Suppose that a club consists of 25 members, and that a manager and a secretary are to be chosen from the membership. The number of all possible choices is
P25,2 = 25 × 24 = 600.
Example 6.3 (Ordered sampling with replacement) Consider the following experiment: A box contains n balls numbered 1, . . . , n. First, one ball is selected at random from the box and its number is noted. This ball is then put back in the box and another ball is selected. As many balls can be selected in this way. This process is called ordered sampling with replacement. The sample space of this experiment will contain all vectors of the form (x1, . . . , xk), where xiis the outcome of the ith selection. The total number of vectors in S is nk. The probability of the event that each of the k balls that are selected will have a different number is
Example 6.4
(The birthday problem) The problem is to determine the probability p that at least two people in a group of k people (2 ≤k ≤365) will have the same birthday. Under appropriate assumptions (independence and uniformity),the probability that all k persons will have different birthdays is P365,k/365k. The probability that at least two of the people will have the same birthday is
7 Combinatorial Methods
7.1 Combinations
Suppose that there is a set of n distinct elements from which it is desired to choose a subset containing k elements (1 ≤k ≤n). The number of different subsets that can be chosen is
The number Cn,k is also denoted by the symbol . When this notation is used, this number is called a binomial coefficient because it appears in the binomial theorem.
Binomial theorem. For all numbers x and y and each positive integer n,
Examples
Example 7.1 (Unordered sampling without replacement) Suppose that a class contains 15 boys and 30 girls, and that 10 students are to be selected for a special assignment. We shall determine the probability that exactly 3 boys will be selected.
The desired probability is
In summary, we have the following table for counting.
Example 7.3 (Unordered sampling with replacement) A restaurant has n items on its menu. During a particular day, k customers will arrive and each one will choose one item. The manger wants to count how many different collections of customer choices are possible without regard to the order in which the choices are made.
Conditional Probability
A major use of probability in statistical inference is the updating of probabilities when certain events are observed. The updated probability of event Aafter we learned that event B has occurred is the conditional probability of A given B. The notation for this conditional probability is P(A |B).
Example 1.1 Rolling Dice Suppose that two dices were rolled and it was observed that the sum T of the two numbers was odd. We shall determine the probability that T was less than 8.
1.1 The multiplication rule for conditional probability
The definition of conditional probability implies the following multiplication rule,
P(AB) = P(B)P(A |B)
Or
P(AB) = P(A)P(B |A).
Example 1.3 Suppose that two balls are to be selected at random, without replacement, from a box containing r red balls and b blue balls. The probability that the first ball will be red and the second ball will be blue is
2 Independent Events
Definition 2.1 Two events are independent if P(AB) = P(A)P(B).
Suppose that P(A) 0 and P(B) 0, then it follows from the definitions of independence and conditional probability that A and B are independent if and only if:
P(A |B) = P(A) and
P(B |A) = P(B),
that is, learning that B (A) occurred does not change the probability of A (B).
Example 2.1 Machine operations. Suppose that two machines 1 and 2 in a factory are operated independently of each other. Let A be the event that machine 1 will become inoperative during a given 8-hour period; let B be the event that machine 2 will become inoperative during the same period; and suppose that P(A) = 1/3and P(B) = 1/4. What is the probability that both machines will become inoperative during the period.
Theorem 2.1 If two events A and B are independent, then the events A and B care also independent.
Definition 2.2 The k events A1, . . . , Akare independent if, for every subset Ai1 , . . . , Aijof j of these events (j = 2, 3, . . . , k), P(Ai1 \ · · · \ Aij ) = P(Ai1 ) · · · P(Aij ).
In particular, in order for three events A, B, and C to be independent, the following four relations must be satisfied:
P(AB) = P(A)P(B)
P(AC) = P(A)P(C)
P(BC) = P(B)P(C)
P(ABC) = P(A)P(B)P(C).
Example 2.3
Suppose that a machine produces a defective item with probability p (0 < p < 1) and produces a nondefective item with probability q = 1 - p. Suppose further that six items produced by the machine are selected at random and inspected, and that the outcomes for these six items are independent. The probability that at least one of the six items will be defective is
p = 1 - P(D1D2D3D4D5D6) = 1 - q 6 ,
The probability that exactly two of six items are defective is:
Example 2.5 Suppose that the probability p that the machine produces a defective item takes one of two values, either 0.01 or 0.4, the first corresponding to normal operation and the sec-ond corresponding to a need for maintenance. Let B1be the event that p = 0.01, and let B2be the event that p = 0.4. If we knew that B1had occurred, then we would proceed under the assumption that the events D1, D2, . . .were independent with P(Di | B1) = 0.01for all i. Let A be the event that we observe exactly two defectives in a random sample of six items. Then
1 Random variables and discrete distributions
Definition 1.1 Consider an experiment for which the sample space is defined by S. A random variable is a function from a sample space into the real numbers. In other words, in a particular experiment a random variable X would be some function that assigns a real number X(s) to each possible outcome s 2 S.
Example 1.1 Consider an experiment in which a coin is tossed 10 times. In this experiment the sample space can be regarded as the set of outcomes consisting of the 210different sequences of 10 heads and tails that are possible, and the random variable X could be the number of heads obtained on the 10 tosses. For each possible sequence s consisting of 10 heads and tails, this random variable would assign a number X(s) equal to the number of heads in the sequence. Thus, if s is the sequence HHTTTHTTTH, then X(s) = 4.
The distribution of a random variable
When a probability distribution has been specified on the sample space of an experiment, we can determine a probability distribution for the possible values of each random variable X. Let Abe a subset of the real line, and let P(X 2 A) denote the probability that the value of X will belong to the subset A. Then P(X 2 A) is equal to the probability that the outcome s of the experiment will be such that X(s) 2 A. In symbols,
P(X 2 A) = P{s :X(s) 2 A}.
Example 1.3
Consider again an experiment in which a coin is tossed 10 times, and let X be the number of heads that are obtained. In this experiment the possible values of X are 0, 1, 2, . . ., 10.
Then:
for x = 0, 1, . . . , 10.
Discrete distributions (Discrete Random Variable)
It is said that a random variable X has a discrete distribution or that X is a discrete random variable if X can take only a finite number k of different values x1, . . . , xkor, at most, an infinite sequence of different values x1, x2, . . ..
Continuous distributions (ContinuousRandom Variable)
Random variables that can take every value in an interval are called continuous random variables.
Probability Distribution Function (abbreviated p.d.f.)
If a random variable Xhas a discrete distribution, the probability distribution function (abbreviated p.f.) of X is defined as the function f such that for every real number x, f(x) = P(X = x). For every point x that is not one of the possible values of X, it is evident that f(x) = 0. Also, if the sequence x1, x2, . . . includes all the possible values of X, then
The uniform distribution on integers
Suppose that the value of a random variable Xisequally likely to be each of the k integers 1, 2, . . . , k. Then the p.d.f. of X is as follows:
The Binomial distribution
Suppose that a certain machine produces a defective item with a probability of p (0 < p < 1) and produces a nondefective item with a probability of q = 1 −p. Suppose further that n independent items produced by the machine are examined, and letX denote the number of these items that are defective. Then the random variable X will have a discrete distribution, and the possible values of X will be 0,1, . . ., n. The probability of X = x is
2 Continuous Distributions
The probability density function It is said that a random variable X has a continuous distribution or that X is a continuous random variable if there exists a nonnegative function f, defined on the real line, such that for every subset A of the real line, the probability that X takes a value in Ais the integral of f over the set A. In this course, we shall be concerned primarily with subsets that are intervals. For an interval (a, b],
The function f is called the probability density function of X. Every p.d.f. f must satisfy the following two requirements:
f(x) ¸ 0, for all x,
and
The uniform distribution on an interval
Let a and b be two given real numbers such that a < b, and consider an experiment in which a point Xis selected from the interval S = {x : a · x · b} in such a way that the probability that X will belong to each subinterval of S is proportional to the length of that subinterval. This distribution of the random variable X is called the uniformdistribution on the interval [a, b].