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Chapter 5. Probability and Statistics

Chapter 5: PROBABILITY AND STATISTICS__________________________________________________UNIT 1: INTRODUCTION.1. VOCABULARY AND GRAMMAR REVIEW:1.1 Vocabulary:

- Complement (n): phần bù- Continuous data (n): dữ liệu liên tục- Discrete data (n): dữ liệu rời rạc- Disjoint (a): rời nhau- Element (n): phần tử - Empty set (n): tập rỗng- Frequency (n): tần suất- Mean (n): giá trị trung bình- Mean absolute deviation (n): độ lệch tuyệt đối trung bình- Mean deviation (n):độ lệch trung bình- Median (n): giá trị ở giữa; trung vị- Mode (n): mẫu- Mutually exclusive events (n): các biến cố loại trừ hỗ tương- Non-mutually exclusive events (n): các biến cố loại trừ không hỗ tương- Population mean (n): Trung bình dân số- Population variance (n): Phương sai dân số- Probability (n): xác suất- Random experiment (n): dãy thí nghiệm ngẫu nhiên- Sample mean (n): Trung bình mẫu- Sample space (n): không gian mẫu- Sample standard deviation (n): độ lệch chuẩn của mẫu. - Sample variance (n): Phương sai mẫu- Set (n): tập hợp- Standard deviation (n): độ lệch chuẩn- Union (n): phần giao- Variance (n): phương sai- Venn diagram (n): biểu đồ Venn

1.2 Grammar review:- Conditional sentences- Relative clause- Adverbial clause- Tenses

1.3 Practice exercises:1.3.1 Put the correct corresponding between column A and column B

A B

a) A Venn diagram 1) Is a collection or group of object.

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b) A set 2) is a diagram which presents the information in groups.

c) The universal set 3) Is the set of elements which are in A or B, or in both.

d) The union of sets A and B 4) Is the set of all objects we are considering.

e) The complement of set A 5) If it is impossible for both of them to occur at the same time.

f) Two events are said to be mutually exclusive

6) Is the set of elements which are not in A.

1.3.2 Fill in the blanks with the correct form of the verb in the bracket:When the data that we gather come in the form of a ____(1)___(to collect) of numbers

then we refer to the data as being discrete. For example, if we were to count the number of drinks served by 20 different hot drinks machines on a college campus then the numbers collected ____(2)___(to form) discrete data.

Despite the fact that the data ___(3)___(to be) discrete, we sometimes record the data as falling within continuous ranges of values. For example, if we ____(4)___(to perform) a survey to record the salary of all the people employed by a particular company and recorded the value of every single pay-packet it would be entirely possible for every recorded number to be different. This ____(5)___(to be) because salary is a particular value on a continuous scale of values. Continuous, that is, to the nearest penny. In such a case it is much more convenient to record an individual salary as being within a continuous range of money values. The following table ____(6)___(to illustrate) this:

Salary ($) (to nearest $) Number of employees

1 -10000 123

10001 – 20000 257

20001 – 30000 96

30001- 74

Here we ____(7)___(to record) the number of employees whose salary falls within defined ranges of money values. This is ____(8)___(to know) as grouping the data. Notice that in this table the groups represent values that are continuous to the nearest $ so that rounding ____(9)___(to have) to be applied. For example, employee whose salary is $10000.49 is recorded in the first group, whereas an employee with a salary of $10000.50 is recorded in the second group1.3.3 Put the words in the correct order to make meaningful sentences:a) The set with no elements is called the empty set. b) The empty set is also regarded as a subset of every other set. c) A set A is finite if it is empty or if it consists of exactly n elements. d) The set S of all possible outcomes of a given experiment is called the sample space.

Note:

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Sample mean: , where are n numerical values of some

sample.

The population mean , where are N numerical values of the

entire population. The median of the data is defined to be the middle value. Let be the sample mean of n values . The difference is called the deviation of the data value about the mean ; it is positive or negative according as is greater or less than . The sample variance is defined as follows:

Sample variance:

Sample standard deviation:

Population variance:

Population standard deviation: 2. READING:2.1 PROBABILITY

All our lives are affected by chance. From simple board games to gambling with money and understanding of probability may improve our chances. If your job was to insure the owner of a new vehicle, would you rather insure a 21 – year – old male driver or 40 – year – old female driver? If you were a famer, and there was a dry winter forecast, would you plant your crops? Will a worker reaching retiring age invest money that will not mature (become variable) for twenty years? All of these decisions would be best made with a knowledge of the most likely outcomes. Therefore, probability is the study of chance, or likelihood.

Hweeying’s bus has been on time 7 out of the last 10 days she has caught it. The

probability of the bus being on time on the next occasion is .

In general Pr (successful event) = .

Note that in the probability formula, the word “successful” refers to the outcome you want to achieve or the event you are interested in. In the above example, the outcomes we consider to be successful are those where the bus is on time. If the bus was always on time,

then the probability would have been , or 1. If the bus was never on time, then the

probability would have been . So the probability of an event occurring is a number

between 0 and 1.



What is the probability of Hweeying’s bus not being on time on the next occasion? This is called the complementary event to the bus being on time, that is the event we were interested in not occurring.

If we let A represent the event “ Hweeying’s bus is on time” then A’, or the complement of A, represents the event “Hweeying’s bus is not on time”.

.

Notice that Pr(A) + Pr(A’) = 1 or Pr(A’) = 1 – Pr(A). Comprehension check:Answer the following questions:

1) According to the text, what is probability?2) What is the characteristic of an event?3) If Hweeying’s bus is always on time, what is the probability of Hweeying’s bus is

not on time?4) According to the text, what is the complementary event to the bus not on time?5) If the probability that you win the game is 0,35, what is the probability that you will

lost?2.2 RANDOM EXPERIMENT

An experiment is any action or sequence of actions performed to find the value or values of some unknown quantity or quantities. For example, an experiment could be conducted to find the average height of group of people. The experiment method would involve measuring the height of everyone in the group and from these measurements, calculating the average height. The sequence of actions in this experiment consists of measuring and calculating, the value of the unknown quantity found being the average height. The unknown quantity whose value we are trying to find is called the result of the experiment and its value, found from an experiment, is called an outcome of the result. By the very nature of an experiment an outcome is unknown before the experiment is performed; a result can be anticipated but its actual value – the outcome – is unknown until the experiment is completed.

Take for example, the simple experiment of tossing a coin. Before the coin is tossed it is not known whether it will show a head or a tail but it is known that it will show one of the other – it will produce a result. We can, therefore, list the possible outcomes of the result as: Head, Tail.

If an ordinary six-sided die were to be rolled the result will be a numbered face showing uppermost. The possible outcomes of that result are then: One, Two, Three, Four, Five, Six.

Any experiment that has a result with more than one possible outcome is referred to as a random experiment and the complete list of possible outcomes is called the sample space of the experiment. The only requirement that is made of the outcomes of a random experiment is that they are mutually exclusive, that is, only one of the possible outcomes can actually occur when the experiment is performed – the result can have only one value.

Comprehension check Answer the following questions:1) What is an experiment? Give two more examples?



2) What is a random experiment? 3) What is the result of an experiment?4) What is sample space?5) How is mutually exclusive experiment?6) What are the differences between random experiment and exclusive experiment?7) What are the possible outcomes, when you roll an ordinary six-sided die?

2.2 PROBABILITY OF OCCURRENCEIn mathematical language the choice, the probability of success is the ratio of the

number of ways in which the trial can succeed to the total number of ways in which the trial can result. Here nothing favors the choice of any particular circle; they are all on the same page, and you are just as likely to cover one as another. If there are 9 balls, five of

which are black, and the others are white, the probability of choosing a black circle is , If

the event choosing the black circle is the success, then the probability of success is .

Similarly, the probability of failure is the ratio of the number of ways in which the trial can fail to the total number of ways in which it can result. If q represents the probability of

failure, in this case . Notice that the sum of probability of success and failure is 1. If

you put your finger on a circle, it is certain to be either black circle or a white one, for no

other kind of circle is present. Thus . The probability that an event will

occur cannot be more than 1. When p = 1, success is a certain. When q = 1, failure is sure.Let S represent the number of ways in which a trial can succeed. And let f represent the

number of ways in which a trial can fail.

When S is greater than f, the odds are S to f in favor of success, thus the odds in favor of covering a black circle are 5 to 4. Similarly, when f is greater than S, the odds are f to S against success. And when S and f are equal, the chances are even; success and failure are equally likely. Tossing a coin illustrates a case in which S and f are equal. There are two sides to a coin, and there is no reason why a normal coin should fall one side up rather than

the other. So if you toss a coin and call heads, the probability that it will fall heads is .

Suppose you toss a coin a hundred times, for each of the hundred trials, the probability that

the coin and call heads, the probability will come down heads is . You might expect fifty

of the tosses to be heads. Of course, you may not get fifty heads. But the more times you toss a coin, the closer you come to the realization of what you expect.

If p is the probability of success on one trial, and K is the number of trials, then the expected number is Kp. Mathematical expectation in this case is defined as Kp.



Comprehension check:1. Answer the following questions:a) What does the article deal with?b) If you were shown 9 red circles and 6 black circles and were asked to choose one of

them which on these circles would be likely to choose? Why?c) Can you give the definition of the probability of failure? What is it?d) What are the odds in case ?e) What are the odds in case f < S?f) Suppose S = f, what would the chances be?g) Could you give some examples to illustrate a case when S and f are equal?2. Are these statements true or false? Correct the false statements.a) The trial can succeed in nine ways when you suppose that you have nine circles.b) The sum of the probability of success and failure is equal to 1.c) The probability that an event will occur can be more than 1.d) In tossing two coins the fact that one fell heads would not affect the way the other fell.

3. SPEAKING – LISTENING – WRITING – DISCUSSION:3.1 Writing:

Find and correct the false in the following paragraphAn event is a subset of the sample space, and differ events may or may not be mutually

exclusive. An event contain a single outcome is called simple event. For example, for the random experiment of tossing a coin, the sample space are: {H, T} and the collection of mutually exclusive subsets is {H} and {T}. They are the only two possible simple events associated with the random experiment.

All random experiments have a unique collect of simple events. However, they are not alway the only events that can be defined. In the random experiment of throwing a die we would define the evens: E1: An even number, E2: A number less than four. These events are then described by the sets: E1={2, 4, 6} E2 = {1, 2, 3} which are not mutually exclusive because the number 2 being common to both.

When two or more random experiments are perform one after another, the final outcomes of the sequence of experiments will consist of combination of the outcomes of the individual experiments. For example, consider the two random experiments of tossing a silver coin followed by tossing a copper coin. We shall denoted the sample space of the first by {SH, ST} and the second by {CH, CT}. Each experiment has two possible outcomes so the sequence of two experiments has four possible combined outcomes. Therefore, the sample space of the random experiment consisting of these two experiments performed in sequence are {(SH, CH), (SH, CT), (ST, CH),(ST, CT)}.

3.2 Listening:Listen to the tape and fill in the blankA set may be viewed as any well-defined _______(1)____ of objects, called the

elements or members of the set. We usually _____(2)____ capital letters; A, B, X, Y … to denote sets, and lower-case letters a, b, x, y, …, to denote _____(3)___ of sets. Synonyms for set are class, collection, and family.



There are essentially _____(4)____ ways to specify a particular set. One way, if possible, is to list its elements. For example, A = {1, 3, 5, 7, 9} means A is the set consisting of the numbers 1, 3, 5, and 9. Note that the element of set _____(5)____separated by commas and enclosed in braces { }. This is _____(6)___ the tabular form or roster method of a set. The _____(7)____ way, called the set-builder form or property _____(8)____, is to state those properties which characterize the element in the set, that is, properties held by the members of the set but not by nonmember. Consider, for example, the expression

B = {x : x is an even integer, x > 0} which is read: “B is the set of x such that x is an even integer and x > 0”. It ______(9)____ the set B whose elements are the positive even integers.

3.3 Discussion:Read the following text. Probability theory is the mathematical modeling of the phenomenon of chance or

randomness. If a coin is tossed in a random manner, it can land heads or tails, but we do not know which of these will occur on a single toss. However, suppose we let s be the number of times heads appears when the coin is tossed n times. As n increases, the ratio f = s/n, called the relative frequency of the outcome, become more stable. If the coin is perfectly balanced, then we expect that the coin will land heads approximately 50 percent of the time or, in other words, the relative frequency will approach ½.

Considering another experiment, the tossing of a six-sided die and observing the number of dots that appear on the top side. Suppose the experiment is repeated n times and let s be the number of times 4 dots appear on the top. Again, as n increases, the relative frequency f = s / n of the outcome 4 becomes more stable. Assuming the die is perfectly balanced, we would expect that the stable or long-run value of this ratio is 1/6, and we say the probability of getting a 4 is 1/6.

Work in group to answer the following questions:- According to the text, what is the probability theory?- What is the relative frequency? Give some more examples experiment and the relative

frequency?- If you toss the coin 100 times, what is the probability of the coin falling head?- If you tossing a six-sided dice 1000 times, what is the probability of the outcome 6?- What is the law of large number? Give example.

4. TRANSLATION:4.1 Translate into Vietnamese:4.1.1Translate these sentences into Vietnamese:

a) Set is a collection of object.b) Element is the individual objects that is contained within a set.c) Empty set is defined as the set that contains no element.d) Mean is obtained by adding a number of data item together and dividing by the

number of data items added.e) Median is the middle datum, when the data are arranged in descending or ascending

order.



f) Mode is the datum that occurs the most in a collection of data. g) The events are said to be mutually exclusive if it is impossible for both of them to

occur at the same time. 4.1.2 Translate into Vietnamese:

All collection of data will have their values spread about the mean value. The mean absolute deviation is a good measure of the spread of the data about the mean as it uses all the data. However, it is very difficult to handle mathematically as opposed to arithmetically. On the other hand, the standard deviation can be analyzed mathematically, so for this reason the standard deviation is the more commonly used measure of spread.

Two sets of data can be compared by comparing their means and spreads. For example, at a Weight Watchers meeting the mean weight of the group and the spread about the mean were measured on two successive weeks. If the two measures of spread were the same but the second mean was less than the first then it could be reasoned that the group as a whole were successfully losing weight. If, however, the two means were the same but the second spread was greater than the first then it could be concluded that some were successful at losing weight while others gained weight, leaving the group’s average weight unaltered.

4.2 Translate into English:Translate each of the following sentences into English:a) Hai tập A và B được gọi là rời nhau nếu chúng không có phần tử chung.Two sets A and B are said to be disjoint if they have no elements in common. b) Phần hợp của hai tập A và B là tập gồm tất cả các phần tử thuộc vào tập A hay thuộc

vào tập B. The union of two sets A and B is the set of all elements which belong to A or to B. c) Phần giao của hai tập A và B là tập gồm các phần tử thuộc vào cả A và B.The intersection of two sets A and B is the set of all elements which belong to both A and B. d) Tập S gồm tất cả các biến cố có thể xảy ra của một thí nghiệm cho trước được gọi là

không gian mẫu. The set S of all possible outcomes of a given experiment is called the sample space.

5. EXERCISES:5.1. Calculate the mean, median and mode of the following data

10, 12, 11, 17, 18, 29, 34, 25, 47, 64,76,10,12, 10.5.2 Find the variance, the standard deviation of the following data, if mean is 30.

10 20 30 40 505.3IN a survey of 50 people, it was found that 36 people like peanut butter while 34

people like Vegemite. Of all those surveyed 20 said they liked both peanut butter and Vegemite. Note P = {people who like peanut butter”, V = “people like Vegemite”.

a) Represent this information on Venn diagram.b) If one person is chosen at random, find: Pr(P); Pr(V); 5.4 A shopkeeper found that, out of his first 20 customers, 13 bought milk and 15

bought a newspaper. Four customers did not buy either milk or a newspaper.a) Represent this information on Venn diagram.



b) How many customers bought both milk and a newspaper?c) Find the probability that a customer, chosen at random from the first 20:- buy milk only- buy milk and a newspaper.- buy milk or a newspaper.5.5 The owner of a small company has 15 employees. Five employees earn $25,000 per

year, seven earn $30,000, three earn $40,000, and the owner’s annual salary is $153,000. Find the mean and the median salaries of all 16 persons in the company.

Puzzles:The Hotel Miramar has two wings, the East Wing and the West Wing. Some East Wing

rooms, but not all, have an ocean view. All West Wing rooms have a harbor view. The charge for all rooms is identical, except as follows:

There is an extra change for all harbor view rooms on or above the third floor.There is an extra charge for all ocean view rooms except those without balcony.Some harbor view rooms on the first two floors and some East Wing rooms without

ocean view have kitchen facilities, for which there is an extra charge. Only the ocean view and harbor view rooms have balconies.Which of the following must be true if all the conditions are as stated?A. All rooms above the third floor involve an extra charge.B. No room without an ocean or harbor view or kitchen facilities involves an extra

charge.C. There is no extra charge for any East Wing room without an ocean view.D. There is no extra charge for any room without kitchen facilities.E. There is an extra charge for all rooms with an ocean or harbor view.

Just for fun: ONE MAN IN BOATDavid: Did you go fishing yesterday? Tom: Yeah, I always go fishing, when I have free time. David: Did you catch anything?Tom: No, I didn’t. I often fish for hours without catching anything, but this does not

worry me. David: It’s a waste of time. Tom: Yeah, I’m not really interested in fishing. I’m interested in sitting in the boat and

doing nothing at all.David: Oh!



Unit 2: PROBABILITY OF SIMPLE EVENT_______________________________________1. VOCABULARY AND GRAMMAR REVIEW:1.1 Vocabulary:

- Bead (n): hạt- Certain event (n): biến cố hiển nhiên- Dependent event (n): biến cố độc lập- Impossible event (n): biến cố không thể xảy ra- Independent event (n): biến cố độc lập- Intersection (n): phần giao- Six-sided die (n): xúc sắc

1.2 Grammar review:- Comparison- Conditional sentences- Gerund and To infinitive

1.3 Practice exercises:1.3.1 Choose the correct word to fill in the blanks:

States defined refuse race unlucky probabilityhas unity impossible

Murphy’s law ___(1)___ that buttered toast always lands butter – side down. As an expression of a law of nature it is a pretty poor one but as an expression of luck and the law of chance it is a good attempt. Chance is ___(2)___ as the absence of design or discoverable cause and many events that we observe seem to happen by chance. Indeed, some people attempt to measure chance in their daily lives, they ___(3)___to walk under a ladder, they throw salt over their left shoulder, they bet money on a horse to win a ___(4)__. In all of these activities there is an in-buit measure of likelihood or probability. If you walk under the ladder you feel that you are unlikely to be___(5)___. If you throw salt over your left shoulder you feel that you are likely to avoid bad luck. If you bet on a horse winning a race what is the ____(6)___of it winning? Who is to know? There are so many factors involved it is impossible to say with certainty. All you can do is to weight the odds – compare the probability of your horse winning against the probability of any other horse winning.

Consider the random experiment of tossing a coin that ____(7)___a head on both sides – a double-headed coin. If you toss this coin then it is certain to fall down heads. The event {H} is a certain event and we define the probability of a certain event as___(8)___:

Pr({H}) = Pr(certainty) = 1Similarly, it is impossible for it to fall down tails – the event {T} is an impossible

event. We define the probability of an ___(9)____event as zero:Pr({T}) = Pr(Impossibility) = 0If the double – headed coin is now replaced by a normal coin possessing a tail as well

as a head then the event {H} is no longer certain and the even {T} is no longer impossible. In each case their probabilities lie between zero and unity:

0 < Pr ({H})<1 and 0 < Pr({H})<1



Assigning correct numbers to these probabilities is not simple but what we do say is that because when a normal coin is tossed it is certain to show either a head or a tail then the two probabilities add up to the probability of certainty, that is unity:

Pr({H}) + Pr({T}) = Pr({certainty}) =12. READING:2.1 PROBABILITY OF SIMPLE EVENT

To every simple event of a random experiment we can assign a measure of the chance of that event occurring. This measure is called the probability of that event occurring.

The probability of a certain event is 1 and the probability of an impossible event is 0. Any other event that is neither certain nor impossible will have a probability between 0 and 1.

Probabilities can be assigned to events a priori – beforehand – or by statistical regularity – afterwards. For example, if we make the assumption that a coin is fair then we have tacitly assigned probabilities beforehand because the notion of fairness means that there is an equal chance of the tossed coin falling down heads or tails. That is: Pr({H}) + Pr({T}) =1 and Pr({H}) = Pr({T}), so that Pr({H}) = Pr({T}) = ½.

Instead of assuming the coin to be fair we tossed it 1000 times and found that we ended up with 510 heads and 490 tails then we would say that, based on statistical regularity, the

probabilities were defined as:

Consequently, probability of a head, Pr({H}) = 510/1000 = 0.51 and Pr({T}) = 490/1000 = 0.49. Thus, the probabilities must add up to 1.

Let E1 and E2 be two events associated with a random experiment. These two events can be connected via the set operation of union to form the event E3, that is . Event E3 can be described in English as either event E1 occurs or event E2 occurs or both occur. This is an inclusive or because it permits both events to occur simultaneously. The probability of E3 occurring depends upon whether E1 and E2 are:

- Mutually exclusive events.- Not mutually exclusive events.If event E1 and E2 are mutually exclusive they contain no outcomes in common. In this

case the union of E1 and E2 is equal to the sum of each probability:

If events E1 and E2 do have outcomes in common, then they are not mutually exclusive. In this case the probability of their union is equal to the sum of the probabilities minus the probability of their intersection:

Comprehension check:Answer the following questions:1) If two events E1 and E2 are not mutually exclusive, what is ? Give

example?2) If two events E1 and E2 are mutually exclusive, what is ? Give example?3) What is the probability of the certainty? What is the probability of the impossibility? 4) What is the characteristic of an arbitrary event?



5) There are 10 balls in a bag, 6 balls are black and 4 balls are white. If you pick a ball, what is the probability of black ball?

3. SPEAKING – LISTENING – WRITING – DISCUSSION:3.1 Discussion:

Work in pair to answer the following questions1) In your opinion, what is probability?2) What is the probability of the certainty?3) What is the probability of your horse to win the race?4) What is the probability of the impossibility?5) If you roll a six-sided die, what is the probability of the side with number six?

3.2 Read the following texta) Translate it into Vietnamese:Let S be a sample space, let be the class of all events, and let P be a real – valued

function defined on . Then P is called a probability function, and P(A) is called the probability of the event A when the following axiom hold:

[P1] For any event A, we have [P2] For the certain event S, we have P(S) = 1.[P3] For any two disjoint events A and B, we have b) Solve the following exercises:1) A basket contains 3 red, 2 white and 5 black scarves. The probability that a scarf

drawn at random will be red or black is:

A: ½ B: 3/10 C: 1 D: 4/5 E: 2/5

2) Twenty gifts numbered in order from 1 to 20 are placed in a bag and one is randomly selected. What is the probability that the number on the token is:

a) 19 or 20b) A multiple of 3c) A multiple of 3 or a multiple of 7d) Even or the number 15e) 5, 6 or 7?

3) In a survey of 60 people, it was found that 40 like lemonade, while 30 people like

cola. Two people said they disliked both.

a) Present this information as a Venn diagram.

b) If one person is chosen at random. Fine the probability that this person:

- Likes lemonade;

- Likes lemonade only;

- Likes cola;

- Likes both lemonade and cola;



4. TRANSLATION:4.1.1 Translate each of the following sentences into Vietnamese:

a) The impossible event has probability zero. b) For any event A, we have c) If A is the subset of B then P(A) is less than or equal to P(B).d) The probability that an event A occurs once E has occurred or, specifically, the

conditional probability of A given E, written P(A|E), is defined as follows:

4.1.2 Translate into VietnameseTHE FUNDAMENTAL LAW OF PROBABILITY

If a certain event can occur in different ways, and if, after it has happened in one of these, another event can occur in different ways, then the ordered pair of events can occur in different ways.

Two random experiments are performed in sequence. Let E1 be an event associated with a first random experiment and E2 be an event associated with a second random experiment. These two events can be connected via the set operation of intersection to form the event E3. . In other words: both event E1 and event E2 occur. The probability of E3

occurring depends upon whether E1 and E2 are: - Dependent events- Independent eventsIf two random experiments are performed in sequence, one after the other, then it may

be possible for the outcome of the first experiment to affect the outcome of the second. If this is the case the outcomes are dependent upon each other and the probabilities change after the first experiment has been performed.

If the outcome of the first experiment does not affect the outcome of the second experiment then the outcomes are independent of each other and the probabilities will not change after the first experiment has been performed.

4.2 Translate into English:Jakob Bernoulli sinh ra vào 6 tháng 1 năm 1654 ở Basel, Switzerland. Lúc trẻ,

Bernoulli học ở trường đại học ở Basel. Theo nguyện vọng của cha, ông lấy bằng về triết học vào năm 1671 và thần học năm năm sau đó. Ông nghiên cứu toán học và thiên văn vào những lúc rãnh rỗi. Đóng góp quan trọng của ông trong toán học là lĩnh vực xác suất. Công trình nghiên cứu của ông về xác suất được trình bày trong luận án của ông, tác phẩm Ars conjectandi. Trong đó, ông trình bày nhiều vấn đề quan trọng, trong đó có “luật số lớn”. 5. EXERCISES:

5.1 Consider the random experiment of drawing a card from a deck of 52 where the result is the suit of the card drawn. What is the sample space of this random experiment? What is the probability of the card which is either Heart or Diamond? What is the probability of the Spade card?

5.2 Consider the random experiment of rolling a six-sided die where the result is the number showing uppermost. What is the sample space?

If we define the two events:

English for mathematics 1


E1: = The number showing is less than 5.E2: = The number showing is even. What are the probabilities of , and ?5.3 A bag contains 7 black beads, 6 white beads and 11 grey beads. A bead is selected

and every bead has an equal probability of being chosen. What are the probabilities of each of the following events

a) E1 = {Black bead} b) E2={White or grey bead} c) E3 = {not white bead}5.4 A bag contains 3 red beads, 5 blue beads and 6 green beads. A bead is selected and

every bead has an equal probability of being chosen. What are the probabilities of each of the following events?

a) E1 = {read bead} b) E2 = {Blue or green bead} c) E3={not a green bead}5.5 What is the probability of selecting male from 100 people where 36 of them are

female if everyone has the same chance of being chosen.5.6 A car park contains 10 red Ford cars, 8 white Ford cars, 12 red Honda cars and 3

white Honda cars. A car is selected at random. What is the probability that it is:a) Either a red car or a Ford.b) Not a white car and not a Hondac) Either a white car or a Honda but not both5.7 A library shelf contains 25 hardback and 48 soft back books. Of the hardback books,

10 are novels, 5 are detective friction and the remainder are text books. Of the soft back books, 15 are text books, 21 are detective friction, and the remainder are novels. A book is selected random. Find the probability that the book selected is:

a) a hardback or a novelb) not a soft back and is not detective frictionc) either hardback or a novel or a textbookd) a novel or a soft back book but not both

5.8 A red die and a blue die are thrown. What is the probability that:a) A red 3 and a blue 5 is displayed.b) Neither a red 1 nor a blue 2 is displayed.

5.9 A car park contains 10 Ford cars and 12 Honda cars. Two cars are stolen at different times, being selected at random. What is the probability that the stolen cars are:

a) A Honda and a Fordb) Both Ford cars.

5.10 A computer lab contains 5 PCs and 6 Apple Macs. A student used one of the computers from noon to 2:00pm and a second student used a computer from 3:00 to 4:00. No one else used a computer at that time. If each student selected the computer to use at random, what is the probability that the computers were:

a) A PCs and an Apple Macb) Both PCsc) How would your answers to (a) and (b) have differed if the two students had been

in the laboratory at the same time.

Puzzles:For a panel of professors to assess the States of the Union Message on public TV, the

producer must choose two Republicans and two Democrats. At least one professor must be



an economist and at least one a military expert. Available Republicans are Abbot, Bartlett, Catlett, Dorset and Everett; available Democrats are Fawcett, Gantlet, Helfet, and Insett. Catlett, Fawcett and Gantlet are economics, Dorset and Insett are military experts. Fawcett will not sit in the same room with Catlett, and will take part only if Abbott is on the panel. Dorset refuses to take part with Gantlet, and Everett refuses to take part with Insett.

If Abbott and Bartlett are chosen as the Republicans, who can be chosen as the Democrats?

A. Fawcett and Insett onlyB. Fawcett and Insett or Gantlet and Insett onlyC. Fawcett and Gantlet or Gantlet and Helfet onlyD. Fawcett and Gantlet or Gantlet and Insett or Helfet and InsettE. Gantlet and Helfet, Gantlet and Insett or Helfet and Insett.

Just for fun: FASTER THAN SOUND!Once a year, a race is held for old cars. A lot of cars entered for this race last year and

there was a great deal of excitement just before it began. One of the most handsome cars was a Roll-Royce Silver Ghost. The most unusual car was a Benz which had only three wheels. Built in 1885, it was the oldest car taking part. After a great many loud explosions, the race began. Many of the cars broke down on the course and some drivers spent more time under their cars than in them! A few cars, however, complete the race. The winning car reached a speed of forty miles an hour – much faster than any of its rivals. It sped downhill at the end of the race and its driver had a lot of trouble trying to stop it. The race gave everyone a great deal of pleasure. It was very different from modern car races but no less exciting.



Unit 3: PROBABILITY DISTRIBUTIONS__________________________________________

1. VOCABULARY AND GRAMMAR REVIEW:

1.1 Vocabulary:

- Random variable (n): biến ngẫu nhiên- Connotation (n): ý nghĩa của một từ- Expectation (n): kỳ vọng- Probability distribution (n): phân bố xác suất- Raffle (n): cuộc xổ số- Random variable (n): biến số ngẫu nhiên- The Binomial probability distribution (n): phân bố xác suất nhị thức- The Normal probability distribution (n): phân bố xác suất chuẩn- The Poisson distribution (n): phân bố Poisson- The Standard Normal probability distribution (n): phân bố xác suất tiêu chuẩn

1.2 Grammar review:

- Relative clause- Conditional sentences- Tenses- Passive – Active voices

1.3 Practice exercises:

1.3.1. Fill in the blanks with the suitable form of the word in the bracket:Every random experiment _____(1)___(give) rise to a ___(2)____(collect) of mutually

exclusive events, each with an associated probability of that event occurring. This collection of probabilities _____(3)____(call) the probability distribution of the random experiment and we have seen how a probability ____(4)_____(distribute) can be divined either from a relative frequency distribution using the notion of _____(5)____(statistic) regularity or from a priori considerations.

Divining the probability distribution using statistical regularity is reasonably straightforward – the random experiment _____(6)_____(repeat) a number of times sufficient to do the job. Divining the probability distribution from a priori ____(7)____(consider) is not always quite straightforward. The process can, however, be ___(8)___(great) assisted when a formalized method __(9)____(be) available but to create such a formalized method requires the notion of a random variable.

Every random experiment ___(10)_____(have) a result and associated with that result is a set of possible events. Because all the simple events are mutually exclusive it is possible to code each simple event with its own unique number.



1.3.2 Fill in the blanks with the suitable form of the word in the bracket:THE BINOMIAL PROBABILITY DISTRIBUTION

The binomial distribution concerns itself with the probability of r successes in n

Bernoulli trials and is ____(1)___(to give) by: , where and

p is the probability of success while q is the probability of failure in each independent trial where p + q = 1.

Each simple event associated with a binomial probability distribution ____(2)___(to be) a sequence of Bernoulli trials and the random variable of a binomial distribution is r – the number of ____(3)___(successful) in a sequence of n Bernoulli trials. The ____(4)___(to expect) of each Bernoulli trial is p-the probability of success. The expectation of a sequence of n Bernoulli trials is the sum of the expectation of each trial:

. Consequently, the expectation of a binomial probability ____(5)___(to

distribute) the mean = E(r) = np. Similarly, the variance is ,

Where pq is the variance of a single Bernoulli trial.1.3.3 Fill in the blanks with the correct preposition:

With of along into to from at___(1)___calculate probabilities ____(2)___ a normally distribution random variable x

requires the calculation of areas under a general normal distribution curve, which is a very involved process. Fortunately, there is a simpler method of calculating these probabilities using a special normal distribution called the standard normal distribution, which has a mean of 0 and a standard deviation of 1.

The standard normal distribution, shown in Fig.2, is a normal distribution ___(3)___ a mean of 0 and a standard deviation of 1, and the random variable ___(4)___the horizontal axis is no longer x but z.

Any normal distribution can be transformed ____(5)___ the standard normal distribution by first shifting the vertical axis across so that it coincides with the mean value

thus changing the horizontal values ____(6)___ x-values to value. Having shifted the vertical axis so that it coincides ____(7)___ the axis of symmetry, the distribution can now be either stretched or squeezed to convert it into a normal distribution with a standard deviation of 1.

If it has to be squeezed it is because the original normal distribution had a standard distribution and if it had to be stretched, the original normal distribution had a standard deviation that was less than 1. In either case the shape ____(8)___ the curve is dictated by the fact that the area beneath it should remain ____(9)___1.

In conclusion, to convert a normal distribution with mean and standard deviation into a standard normal distribution the x-values of the original normal distribution are converted ____(10)___ z-values for the standard normal distribution by the transformation

.



2. READING2.1 Reading: BERNOULLI TRIALS

A Bernoulli trial is any random experiment whose result whose result gives rise to only two possible simple events which we shall call {success} with probability p and {failure} with probability q. Pr ({success}) = p and Pr ({failure}) = q. where, naturally: p + q = 1.

A typical Bernoulli trial is the tossing of a coin where a head could be considered as success and a tail as failure. The actual connotations of success and failure are unimportant. What is important is that the random experiment has only two possible outcomes.

Sometimes, we shall concerned not with just a single Bernoulli trial but rather with a succession of such trials where each successive trial is independent of the previous trials, and the probability of r successes in n trials, Pr(r:n) is .

For a single Bernoulli trial there are two simple event, namely (success) and (failure). The probabilities of these two events are

Pr(0:1) = q, the probability of {failure}Which occurs when the random variable r = 0Pr(1:1) = p, the probability of {success}Which occurs when the random variable r = 1. Consequently, the expectation of r for a

single Bernoulli trial is: .

Similarly, the variance is .

The mean of a single Bernoulli trial is p and the variance is pq.Comprehension check:Answer the following questions:

1) What is a Bernoulli trial? Give example?2) What is the formula of the probability of s successes in n trials?

What is the formula of the expectation for a Bernoulli trial?2.2 RARE EVENTS AND THE POISSON DISTRIBUTION

A rare event is one whose probability is very small. For example, a roulette wheel displays 37 numbers from 0 to 36 and against each number is a slot for a rolling ball to fall into. The wheel is spun and the ball is rolled. When the wheel stops the ball comes to rest alongside a number and if you had placed a bet against this number you would have won. Assuming that in a single spin of the wheel each number has an equal chance of being selected by the ball when the probability of {success} – the probability that your selection comes up – is: p = 1/37 = 0.027 which is sufficiently small to be considered a rare event. For rare events it is found that the Poisson probability distribution is a reasonable approximation to the binomial probability distribution and much easier to evaluate. The Poisson probability distribution is given as:

where - the mean of the binomial probability distribution. To maintain a reasonable approximation to the binomial distribution we must ensure that . For example, to compute the chances of winning once in 10 plays of roulette where the probability of {success} p is 0.027. We have, n = 10, p = 0.027 so .



As a consequence the probability of winning once in 10 plays is given to a good approximation by the Poisson distribution as: , and the probability of winning twice in 10 plays is .

Comprehension check:Choose True (T) or False (F). Correct the false1) _______ A rare event is the event that seldom happens.2) _______For rare event, it’s necessary to use the Poisson probability distribution.3) _______The Poisson probability distribution is easier to evaluate than the binomial

probability distribution. 4) _______ The formula of the Poisson probability distribution is

where

2.3 THE NORMAL DISTRIBUTION

The binomial and the Poisson probability distributions are examples of discrete probability distributions – they give the probabilities associated with the values of a discrete random variable r.

While many random experiments such as tossing coins or rolling dice can be described using a discrete random variable, many others cannot. For example, if the random experiment involved measuring the height of a person the result of the experiment would be a measured height. This result would then have to be described using a continuous variable because height is a continuous quantity – no individual can grow from 1m to 2m in height without attaining a continuous scale of heights in between. It does not matter that the recorded measure might be discrete, the value of the result itself must be considered as being available on a continuous scale of values.

For a continuous random variable to have with it associated probabilities, the probability distribution itself must necessarily be continuous. The most useful and most common of these continuous probability distributions is the normal or Gaussian probability distribution.

In the case of a discrete probability distribution each value of the random variable has an associated probability and the sum of all such probabilities is unity. For a continuous probability distribution we cannot associate a probability to a given value of a random variable because there is an infinity of such values and hence there would be an infinity of associated probabilities whose sum would exceed unity. Instead we associate probabilities to ranges of values of random variable, where the probabilities are equal to the area beneath the graph of the probability distribution within that range. For example, in the graph of the normal


x1x2

P(x)

x

area A

Fig.1


probability distribution of the random variable x shown in fig.1, the probability that x is found to have a value less than x2 but greater than x1 is equal to the area A: P(x1 < x < x2) = A. Notice that this probability is the same as and

. The addition of single points at the ends of the range does not affect the probability.

Comprehension check:Answer the following questions:1) What is the characteristic of the discrete probability distributions? Give examples?2) What is the difference between the discrete probability distributions and the

continuous probability distributions?3) What is the characteristic of the continuous probability distributions? Give

examples?4) What is the probability of the random variable x which is greater than or equal to x 1

and less than x2 in the fig.1?Writing: Write a short paragraph about the characteristics of the Normal or Gaussian Probability distribution and the Standard Normal Distribution.3. SPEAKING – WRITING – LISTENING - DISCUSSION3.1. Writing

A. Find the false and correct it:EXPECTATION

Every permitted value of a random variable x associated with a random experiment have a probability Pr(x) of being realized.

In complete analogy with defining the average values of a collect of data as the sum of the product of the various datum values with the relative frequencies so we can defined an average value of a random variable as the sum of the products of its values with their respective probabilities. (the symbol is a lower case Greek m, pronounce “myou”). Here, the average random variable is called the expectation of x, denoted by E(x). That is the expectation of x. Note that the use of the word “expectation” can be misleading. It don’t mean that the most likely value to be expected is , because the most likely value to be expected is that with the higher probability.

B. Base on the text, answer the following questions:

- What is the expectation of the random variable x?

- Is the expectation of x is the most likely value to be expected? Why or Why not?

- What are the applications of the expectation of the random variable x?

3.2 Discussion:

3.2.1) Consider the five successive tossing of a single coin – the fivefold repetition of a single random experiment where the result of each repetition is independent of the result of its predecessor. One such sequence of outcomes could be: H, H, T, T, T. That are 2 heads and 3 tails. Let E be the event associated with the outcome is:



. What is the probability of event E?

3.2.2) A production line produced items of which 1% are defective. If a sample of 10 is

taken from the line what is the probability that:

(a) One item is defective.

(b) All items are perfect.

(c) No more than two items are defective.

3.2.3) A weekly raffle is held between 100 members of a club. There is only one winner

each week and the draw is fair. Find the probability of one individual member winning:

a) Once in 52 weeks.

b) More than 2 times in 52 weeks.

3.3 Writing:

Write a short paragraph about the similarity and the difference between the Bernoulli

probability distribution and the Poisson probability distribution.

4. TRANSLATION:

4.1 Translate into English:

Xác suất của biến cố A bằng số khả năng xảy ra của biến cố A chia cho tổng số các khả

năng có thể xảy ra. Xác suất của biến cố A được ký hiệu là P(A) và nhận một số giữa 0 và

1. Nếu P(A) > P(B) thì biến cố A có khả năng xảy ra nhiều hơn biến cố B. Nếu P(A) =

P(B) thì cả hai biến cố A và B đều có khả năng xảy ra như nhau.

4.2 Translate into Vietnamese:

Just as with discrete probability distributions where the sum of probabilities is unity,

with a continuous probability distribution, the area under the curve is unity. Thus the

probability that a variable has a value lying within a given range of values is given by the

area under the curve between two extremes of the range of values. In fig.1, the probability

that is equal to the area A. As can be seen from the graph, the bell-shaped curve

rising from small P(x) to a maximum and then falling to small P(x) again, is symmetric

about a vertical line through the maximum. Where this vertical line meets the x axis is the

mean of the normal distribution



5. Exercises:

5.1) Two dice are thrown. If you score a 7 or an 11 then you win, otherwise you lose.

Define a random variable x whose values denote a win or a lost. Find both the mean and

standard deviation of this random variable.

5.2) A company has found that on average 20% of its 50 accounts are outstanding at the

end of month. What is the probability that at the end of a particular month:

a) ten accounts are outstanding

b) fifty accounts are outstanding

Puzzles:Joe, Larry, Ned, Mary, Paul, Willy, Crystal, Albert, Bob, Frank, Ellen and Rick all live

in the same six-floor building. There are two apartments per floor. No more than two persons live in any apartment. Some apartments may be empty.

Larry and his roommate live two floors above Albert and his roommate, Crystal.Joe lives alone, three floor below Willy and two floors below Ellen.Mary lives one flow below Albert and CrystalNed lives three floors above the floor in which Bob and Frank have single apartments.

Rick and Paul live in single apartment two floors below Mary. Which of the following lists the persons named in the correct order, going from the

top to the bottom floor?A. Rick, Bob, Mary, Albert, Larry, Ned.B. Rick, Frank, Ned, Ellen, Larry, CrystalC. Paul, Bob, Joe, Crystal, Ned, LarryD. Larry, Ellen, Albert, Mary, Frank, RickE. Larry, Joe, Mary, Albert, Bob, Rick.



REFERENCES:

Bernard Kolman – David R.Hill. Elementary Linear Algebra. Pearson education

international.

Ho Thi Phuong – Le Thi Kieu Van. English for mathematics. Ho Chi Minh City

University of Education. 2003

Reinout Quispel. Introduction to number theory. 2006

Trương Văn Hung – Ts. Nguyễn Phi Khứ. Hệ thống kiến thức toán học & Anh ngữ để

theo học các trường đại học quốc tế. NXB. Giao thông Vận tải. 2006.

V.V.Zaitsev – V.V. Ryzhkov – M.I. Skanavi. Elementary Mathematics. Mir Publisher

Moscow.

Note: The lecture is available on the website: http://www.englishformaths.wikispaces.com


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