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CHAPTER 1
RANDOM EVENTS
1. RELATIONS AMONG RANDOM EVENTS
Basic Formulas
Random events are usually designated by the letters A, B, C, ..., U, V, where U denotes an event certain to occur and V an impossible event. The equality A = B means that the occurrence of one of the events inevitably brings about the occurrence of the other. The intersection of two events A and B is defined as the event C = AB, said to occur if and only if both events A and B occur. The union of two events A and B is the event C = A\B said to occur if and only if at least one of the events A and B occurs. The difference of two events A and B is defined as the event , said to occur if and only if A occurs and B does not occur. The complementary event is denoted by the same letter as the initial event, but with an over bar. For instance, [bar.A] and A are complementary, [bar.A] meaning that A does not occur. Two events are said to be mutually exclusive if AB = V. The events Ak (k = 1, 2, ..., n) are said to form a complete set if the experiment results in at least one of these events so that [MATHEMATICAL EXPRESSION OMITTED].
Solution for Typical Examples
Example 1.1 What kind of events A and B will satisfy the equality A [union] B = A?
Solution. The union A [union] B means the occurrence of at least one of the events A and B. Then, for A [union] B = A, the event A must include the event B. For example, if A means falling into region SA and B falling into region SB, then SB lies within SA.
The solution to Problems 1.1 to 1.3 and 1.8 is similar.
Example 1.2 Two numbers at random are selected from a table of random numbers. If the event A means that at least one of these numbers is prime and the event B that at least one of them is an even number, what is the meaning of events AB and A [union] B?
Solution. Event AB means that both events A and B occur. The event A [union] B means that at least one of the two events occurs; that is, from two selected numbers at least one number is prime or one is even, or one number is prime and the other is even.
One can solve Problems 1.4 to 1.7 analogously.
Example 1.3 Prove that [MATHEMATICAL EXPRESSION OMITTED] and [MATHEMATICAL EXPRESSION OMITTED].
Proof. If C = [bar.A] and D = [bar.B], the second equality can be written in the form [bar.A [union] B]] . Hence it suffices to prove the validity of the first equality.
The event [bar.A][bar.B] means that both events A and B do not occur. The complementary event [MATHEMATICAL EXPRESSION OMITTED] means that at least one of these events occurs: the union A [union] B. Thus [MATHEMATICAL EXPRESSION OMITTED]. The proof of this equality can also be carried out geometrically, an event meaning that a point falls into a certain region.
One can solve Problem 1.9 similarly. The equalities proved in Example 1.3 are used in solving Problems 1.10 to 1.14.
Example 1.4 The scheme of an electric circuit between points M and N is represented in Figure 1. Let the event A be that the element a is out of order, and let the events Bk (k = 1, 2, 3) be that an element bk is out of order. Write the expressions for C and [bar.C] where the event C means the circuit is broken between M and N.
Solution. The circuit is broken between M and N if the element a or the three elements bk (k = 1, 2, 3) are out of order. The corresponding events are A and B1B2B3. Hence C = A [union] B1B2B3.
Using the equalities of Example 1.3, we find that
[MATHEMATICAL EXPRESSION OMITTED]
Similarly one can solve Problems 1.16 to 1.18.
PROBLEMS
1.1 What meaning can be assigned to the events A [union] A and A A?
1.2 When does the equality AB = A hold?
1.3 A target consists of 10 concentric circles of radius rk (k = 1, 2, 3, ..., 10). An event Ak means hitting the interior of a circle of radius rk (k = 1, 2, ..., 10). What do the following events mean:
[MATHEMATICAL EXPRESSION OMITTED]
1.4 Consider the following events: A that at least one of three devices checked is defective, and B that all devices are good. What is the meaning of the events (a) A [union] B (b) AB?
1.5 The events A, B and C mean selecting at least one book from three different collections of complete works; each collection consists of at least three volumes. The events As and Bk mean that s volumes are taken from the first collection and k volumes from the second collection. Find the meaning of the events (a) A [union] B [union] C (b) ABC, (c) A1 [union] B3, (d) A2B2, (e) (A1B3B1A3)C.
1.6 A number is selected at random from a table of random numbers. Let the event A be that the chosen number is divisible by 5, and let the event B be that the chosen number ends with a zero. Find the meaning of the events A\B and [bar.AB].
1.7 Let the event A be that at least one out of four items is defective, and let the event B be that at least two of them are defective. Find the complementary events [bar.A] and [bar.B].
1.8 Simplify the expression [MATHEMATICAL EXPRESSION OMITTED].
1.9 When do the following equalities hold true: (a) A [union] B = [bar.A], (b) AB = [bar.A], (c) A [union] B = AB?
1.10 From the following equality find the random event X:
[MATHEMATICAL EXPRESSION OMITTED]
1.11 Prove that [MATHEMATICAL EXPRESSION OMITTED]
1.12 Prove that the following two equalities are equivalent:
[MATHEMATICAL EXPRESSION OMITTED]
1.13 Can the events A and [bar.A [union] B] be simultaneous?
1.14 Prove that A, [bar.A]B and [bar.A [union] B] form a complete set of events.
1.15 Two chess players play one game. Let the event A be that the first player wins, and let B be that the second player wins. What event should be added to these events to obtain a complete set?
1.16 An installation consists of two boilers and one engine. Let the event A be that the engine is in good condition, let Bk(k = 1, 2) be that the kth boiler is in good condition, and let C be that the installation can operate if the engine and at least one of the boilers are in good condition. Express the events C and in terms of A and Bk.
1.17 A vessel has a steering gear, four boilers and two turbines. Let the event A be that the steering gear is in good condition, let Bk (k = 1, 2, 3, 4) be that the boiler labeled k is in good condition, let Cj (j = 1, 2) be that the turbine labeled j is in good condition, and let D be that the vessel can sail if the engine, at least one of the boilers and at least one of the turbines are in good condition. Express D and [bar.D] in terms of A and Bk.
1.18 A device is made of two units of the first type and three units of the second type. Let Ak (k = 1, 2) be that the kth unit of the first type is in good condition, let Bj (j = 1, 2, 3) be that the jth unit of the second type is in good condition, and let C be that the device can operate if at least one unit of the first type and at least two units of the second type are in good condition. Express the event C in terms of Ak and Bj.
CHAPTER 2
A DIRECT METHOD FOR EVALUATING PROBABILITIES
Basic Formulas
If the outcomes of an experiment form a finite set of n elements, we shall say that the outcomes are equally probable if the probability of each outcome is 1/n. Thus if an event consists of m outcomes, the probability of the event is p = m/n.
Solution for Typical Examples
Example 2.1 A cube whose faces are colored is split into 1000 small cubes of equal size. The cubes thus obtained are mixed thoroughly. Find the probability that a cube drawn at random will have two colored faces.
Solution. The total number of small cubes is n = 1000. A cube has 12 edges so that there are eight small cubes with two colored faces on each edge. Hence m = 12·8 = 96, p = m/n = 0.096.
Similarly one can solve Problems 2.1 to 2.7.
Example 2.2 Find the probability that the last two digits of the cube of a random integer will be 1.
Solution. Represent N in the form N = a + 10b + ···, where a, b, ... are arbitrary numbers ranging from 0 to 9. Then N3 = a3 + 30a2b + ···. From this we see that the last two digits of N3 are affected only by the values of a and b. Therefore the number of possible values is n = 100. Since the last digit of N is a 1, there is one favorable value a = 1. Moreover, the last digit of (N3 – 1)/10 should be 1; i.e., the product 3b must end with a 1. This occurs only if b = 7. Thus the favorable value (a = 1, b = 7) is unique and, therefore, p = 0.01.
Similarly one can solve Problems 2.8 to 2.11.
Example 2.3 From a lot of n items, k are defective. Find the probability that / items out of a random sample of size m selected for inspection are defective.
Solution. The number of possible ways to Choose m items out of n is Cmn. The favorable cases are those in which l defective items among the k defective items are selected (this can be done in Clk ways), and the remaining m – l items are nondefective, i.e., they are chosen from the total number n – k (in [MATHEMATICAL EXPRESSION OMITTED] ways). Thus the number of favorable cases is [MATHEMATICAL EXPRESSION OMITTED]. The required probability will be [MATHEMATICAL EXPRESSION OMITTED].
One can solve Problems 2.12 to 2.20 similarly.
Example 2.4 Five pieces are drawn from a complete domino set. Find the probability that at least one of them will have six dots marked on it.
Solution. Find the probability q of the complementary event. Then p = 1 – q. The probability that all five pieces will not have a six (see Example 2.3) is [MATHEMATICAL EXPRESSION OMITTED] and, hence,
[MATHEMATICAL EXPRESSION OMITTED]
By a similar passage to the complementary event, one can solve Problems 2.21 and 2.22.
PROBLEMS
2.1 Lottery tickets for a total of n dollars are on sale. The cost of one ticket is r dollars, and mof all tickets carry valuable prizes. Find the probability that a single ticket will win a valuable prize.
2.2 A domino piece selected at random is not a double. Find the probability that the second piece also selected at random, will match the first.
2.3 There are four suits in a deck containing 36 cards. One card is drawn from the deck and returned to it. The deck is then shuffled thoroughly and another card is drawn. Find the probability that both cards drawn belong to the same suit.
2.4 A letter combination lock contains five disks on a common axis. Each disk is divided into six sectors with different letters on each sector. The lock can open only if each of the disks occupies a certain position with respect to the body of the lock. Find the probability that the lock will open for an arbitrary combination of the letters.
2.5 The black and white kings are on the first and third rows, respectively, of a chess board. The queen is placed at random in one of the free squares of the first or second row. Find the probability that the position for the black king becomes checkmate if the positions of the kings are equally probable in any squares of the indicated rows.
2.6 A wallet contains three quarters and seven dimes. One coin is drawn from the wallet and then a second coin, which happens to be a quarter. Find the probability that the first coin drawn is a quarter.
2.7 From a lot containing m defective items and n good ones, s items are chosen at random to be checked for quality. As a result of this inspection, one finds that the first k of s items are good. Determine the probability that the next item will be good.
2.8 Determine the probability that a randomly selected integer N gives as a result of (a) squaring, (b) raising to the fourth power, (c) multiplying by an arbitrary integer, a number ending with a 1.
2.9 On 10 identical cards are written different numbers from 0 to 9. Determine the probability that (a) a two-digit number formed at random with the given cards will be divisible by 18, (b) a random three- digit number will be divisible by 36.
2.10 Find the probability that the serial number of a randomly chosen bond contains no identical digits if the serial number may be any five-digit number starting with 00001.
2.11 Ten books are placed at random on one shelf. Find the probability that three given books will be placed one next to the other.
2.12 The numbers 2, 4, 6, 7, 8, 11, 12 and 13 are written, respectively, on eight indistinguishable cards. Two cards are selected at random from the eight. Find the probability that the fraction formed with these two random numbers is reducible.
2.13 Given five segments of lengths 1, 3, 5, 7 and 9 units, find the probability that three randomly selected segments of the five will be the sides of a triangle.
2.14 Two of 10 tickets are prizewinners. Find the probability that among five tickets taken at random (a) one is a prizewinner, (b) two are prizewinners, (c) at least one is a prizewinner.
2.15 This is a generalization of Problem 2.14. There are n + m tickets of which n are prizewinners. Someone purchases k tickets at the same time. Find the probability that s of these tickets are winners.
2.16 In a lottery there are 90 numbers, of which five win. By agreement one can bet any sum on any one of the 90 numbers or any set of two, three, four or five numbers. What is the probability of winning in each of the indicated cases ?
2.17 To decrease the total number of games, 2n teams have been divided into two subgroups. Find the probability that the two strongest teams will be (a) in different subgroups, (b) in the same subgroup.
2.18 A number of n persons are seated in an auditorium that can accommodate n + k people. Find the probability that m ≤ n given seats are occupied.
2.19 Three cards are drawn at random from a deck of 52 cards. Find the probability that these three cards are a three, a seven and an ace.
2.20 Three cards are drawn at random from a deck of 36 cards. Find the probability that the sum of points of these cards is 21 if the jack counts as two points, the queen as three points, the king as four points, the ace as eleven points and the rest as six, seven, eight, nine and ten points.
2.21 Three tickets are selected at random from among five tickets worth one dollar each, three tickets worth three dollars each and two tickets worth five dollars each. Find the probability that (a) at least two of them have the same price, (b) all three of them cost seven dollars.
2.22 There are 2n children in line near a box office where tickets priced at a nickel each are sold. What is the probability that nobody will have to wait for change if, before a ticket is sold to the first customer, the cashier has 2m nickels and it is equally probable that the payments for each ticket are made by a nickel or by a dime.
3. GEOMETRIC PROBABILITIES
Basic Formulas
The geometric definition of probability can be used only if the probability of hitting any part of a certain domain is proportional to the size of this domain (length, area, volume, and so forth), and is independent of its position and shape.
If the geometric size of the whole domain equals S, the geometric size of a part of it equals SB, and a favorable event means hitting SB, then the probability of this event is defined to be
p = SB/S
The domains can have any number of dimensions.
(Continues…)
Excerpted from "Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions"
by .
Copyright © 1968 Dr. Richard A. Silverman.
Excerpted by permission of Dover Publications, Inc..
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