RB Astillero
Probability Theory
Probability of Simple Events
PROBABILITY OF SIMPLE EVENTS STATISTICAL EXPERIMENTS Statisticians use the word experiment to describe any process that generates a set of data. A simple example of a statistical experiment is the tossing of a coin. In this experiment there are only two possible outcomes, heads or tails. Another experiment might be conducting a survey on the reaction of tax payers on the new expanded VAT. The opinion of the tax payers concerning e-VAT can also be considered as observations of an experiment. We are particularly interested in the observations obtained by repeating the experiment several times. In most cases the outcomes will depend on chance and, therefore, cannot be predicted with certainty. If a chemist runs an analysis several times under the same conditions, he will obtain different measurements, indicating an element of chance in the experimental procedure. Even when a coin is tossed repeatedly, we cannot certain that a given toss will result in a head. However, we know the entire set of possibilities for each toss. SAMPLE SPACE Definition The set of all possible outcomes of a statistical experiment is called the sample space and is represented by the symbol or S.
Each outcome in the sample space is called element or member of the sample space, or simply a sample point. Examples: 1. Experiment: tossing of a coin = {H, T} Where = the sample space H = head sample points T = tail 2. Experiment: tossing a die Let S1 = the numbers that show on the top face S1 = {1, 2, 3, 4, 5, 6} Let S2 = the number is either even or odd S2 = {even, odd} 3. An experiment consists of flipping a coin and then flipping it a second time if a head occurs. If a tail occurs on the first flip, then a die is tossed once. List the elements of the sample space. = {HH, HT, T1, T2, T3, T4, T5, T6} 4. Three items are selected at random from a manufacturing process. Each item is inspected and classified as defective, D, or nondefective, N. List the elements of the sample space. S= {DDD, DDN, DND, DNN, NDD, NDN, NND, NNN}
1
RB Astillero
Probability Theory
Probability of Simple Events
EVENTS For any given experiment we may be interested in the occurrence of certain events rather than the outcome of a specific element in the sample space. For instance, we may be interested in the event A that the outcome when a die is tossed is divisible by 3. This will occur if the outcome is an element of the subset A = {3, 6} of the sample space = {1, 2, 3, 4, 5, 6} Definition An event is a subset of a sample space. AS Examples 1. Given the sample space S = {t | t ≥ 0}, where t is the life in years of a certain electronic th component, then the the event A that the component component fails before the end of the 5 year is the subset A = {t | 0 t < 5} 2. In the manufacturing process where 3 items are selected at random and classified as D for defective and N for nondefective, let B be the event that the number of defectives is greater than 1. Find the elements of B. B= {DDN, DND, NDD, DDD} OPERATIONS WITH EVENTS Since events and sample space are sets then their relationships and operations can be carried out using set theory.
1. Complement of an Event Definition The complement of an event A , denoted by the symbol A’, with respect to S is the subset of all elements of S that are not in A. Example Let R be the event that a red card is selected from an ordinary deck of 52 playing cards, and let S be the entire deck. Then R’ is the event that the card selected is black.
R
R’
R’ is shaded 2. Intersection of Events Definition The intersection of two events A and B, denoted by the symbol A B, is the event containing all elements that are common to A and B.
2
RB Astillero
Probability Theory
Probability of Simple Events
Example Let A be the event that a person selected at random while dining at a popular cafeteria is a taxpayer, and let B be the event that the person is over 65 years of age. Then the event A B is the set of all taxpayers in the cafeteria who are over 65 years of age.
A B is shaded 3. Mutually Exclusive Events Definition Two events A and B are mutually exclusive, or disjoint if A B = , that is, if A and B have no elements in common. Example A cable television company offers programs of 8 different channels, 3 of which are affiliated with ABC, two with NBC, and 1 with CBS. The other 2 are an educational channel and the ESPN sports channel. Suppose that a person subscribing to this service turns on a television set without first selecting the channel. Let A be the event that the program belongs to the NBC network and B the event that it belongs to the CBS network. Since a television program cannot belong to more than one network, the events A and B have no programs in common. Therefore, the intersection A B contains no programs, and consequently the events A and B are mutually exclusive. 4. Union of events Definition The union of two events A and B, denoted by the symbol A B, is the event containing all the elements that belong to A and B or both. Example Let A be the event that an employee selected at random from oil drilling company smokes cigarettes. Let B be the event that the employee selected drinks alcoholic beverages. Then the event A B is the set of all employees who either drink or smoke, or do both.
A B is shaded 3
RB Astillero
Probability Theory
Probability of Simple Events
Exercises 3.1 1. An experiment consists of tossing a die and then flipping a coin once if the number on the die is even. If the number on the die is odd, the coin is flipped twice. Using the notation 4H, for example, to denote the outcome that the die comes up 4 and then the coin comes up heads, and 3HT to denote the outcome that the die comes up 3 followed by a head and then a tail on the coin, determine the elements of the sample space S. 2. Two jurors are selected from 4 alternatives to serve at a murder trial. Using the notation A1A3, for example, to denote the simple event that alternates 1 and 3 are selected, list the 6 elements of the sample space S. 3. Four students are selected at random from a chemistry class and classified as male or female. List the elements of the sample space S1 using the letter M for male and F for female. Define a second sample space S2 where the elements represent the number of females selected. 4. An experiment involves tossing a pair of dice, 1 green and 1 red, and recording the numbers that come up. (a) If x equals the outcome on the green die and y the outcome on the red die, describe the sample space S. (b) List the elements corresponding to the event A that the sum is greater than 8. (c) List the elements corresponding to the event B that a 2 occurs on either die. (d) List the elements corresponding to the event C that a number greater than 4 comes up on the green die. 5. An engineering firm is hired to determine if certain waterways are safe for fishing. Samples are taken from three rivers. (a) List the elements of a sample space S using the letters F for “safe to fish” and N for “not safe to fish”. (b) List the elements of S corresponding to the event E that at least two of the rivers are not safe for fishing. (c) Define the event that has as its elements the points {FFF, NFF, FFN, F FN, NFN}. 6. The resume’ resume’s of 2 male applicants for a college teaching position in chemistry are placed in the file as the resume’ resume’s of 2 female applicants. Two positions become available and the first, at the rank of assistant professor, is filled by selecting 1 of the 4 applicants at random. The second position, at the rank of instructor, is then filled by selecting at random 1 of the remaining 3 applicants. Using the notations M 2F1, for example, to denote the simple event nd that the first position is filled by the 2 male applicant and the second position is filled by the first female applicant: (a) list the elements of the sample space S; (b) list the elements of S corresponding to event A that the position of assistant professor is filled by a male applicant; (c) list the elements of S corresponding to event B that exactly 1 of the 2 positions was filled by a male applicant; (d) list the elements of S corresponding to event C that neither position was filled by a male applicant; (e) list the elements of S corresponding to the event A B; (f) list the elements of S corresponding to the event A C; (g) construct a Venn diagram to illustrate the intersections and unions of the events A, B, and C. 7. Consider the sample space = {copper, sodium, nitrogen, potassium, uranium, oxygen, zinc} and the events A = {copper, sodium, zinc}, B = {sodium, nitrogen, potassium}, and C = {oxygen}. List the elements of the sets corresponding to the following events: (a) A’ (d) B’ C’ (b) A C (e) A B C (c) (A B’) C’ (f) (A’ B’) (A’ (A’ C)
4
RB Astillero
Probability Theory
Probability of Simple Events
COUNTING SAMPLE POINTS One of the challenges that statisticians have to deal with when performing statistical experiments is counting the number of points in the sample space without actually listing each element. This task is easily addressed with the use of combinatorics techniques. Examples 1. A president and a treasurer are to be chosen from a student club consisting of 50 people. How many different choices of officers are possible if (a) there are no restrictions; (b) A will serve only if he is president; (c) B and C will serve together or not at all; and (d) D and E will not serve together. Solutions: (a) N = 50P2 = 50!/48! = 2450 (b) N = 49 + 49P2 = 49 + 2352 = 2401 (c) N = 2 + 48P2 = 2 + 2256 = 2258 (d) N = 2(96) + 48P2 = 2448 2. A boy asks his mother to get five game-boy cartridges from his collection of 10 arcade and 5 sports game. How many ways are there that his mother will get 3 arcade and 2 sports games, respectively? Solution: N = 10C3 x 5C2 = 120 x 10 = 1200 ways
Exercises 3.2 1. In a medical study patients are classified in 8 ways according to whether they have blood + + + + type AB , AB , A , A , B , B , O , or O , and also according to whether their blood pressure is low, normal, or high. Find the number of ways i n which a patient can be classified. Ans. 24 2. If an experiment consists of throwing a die and then drawing a letter at random from the English alphabet, how many points are in the sample space? Ans. 156 3. Students at a private liberal arts college are classified as being freshmen, sophomores, juniors, or seniors, and also according to whether they the y are male or female. Find the total number of classifications for the students of that college. Ans. 8 4. A California study concluded that by following 7 simple health rules a man’s life can be extended by 11 years on the average and a woman’s life by 7 years. yea rs. These 7 rules are: no smoking, regular exercise, use alcohol moderately, get 7 to 8 hours of sleep, maintain proper weight, eat breakfast, and do not eat between meals. In how many ways can a person adopt 5 of these rules to follow: (a) If the person presently violates all 7 rules? (b) If the person never drinks and always eats breakfast? Ans. 21, 1 5. A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different plans are available to this buyer? Ans. 48
5
RB Astillero
Probability Theory
Probability of Simple Events
6. A drug for the relief of asthma can be purchased from 5 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. How many different ways can a doctor prescribe the drug for a patient suffering from asthma? Ans. 30 7. In a fuel economy study, each 3 race cars is tested using 5 different brands of gasoline at 7 test sites located in different regions of the country. If 2 drivers are used in the study, and test runs are made once under each distinct set of conditions, how many test runs are needed? Ans. 210 8. A witness to a hit-and-run accident told the police that the license number contained the letters RLH followed by 3 digits, the first of which is a 5. If the witness cannot recall the last 2 digits, but is certain that all 3 digits are different, find the maximum number of automobile registrations that the police may have to check. Ans. 72 9. (a) In how many ways can 6 people be lined up to get on a bus? (b) If 3 specific persons insist on following each other, how many ways are possible? (c) If two specific persons refuse to follow each other, how many ways are possible? Ans. 720, 144, 480 10. A contractor wishes to build 9 houses, each each different in design. In how many ways can he place these houses on a street if 6 lots are on one side of the street and 3 lots are on the opposite side? Ans. 362, 880 10. Four married couples have bought 8 seats in the same row for a concert. In how many different ways can they be seated: (a) With no restrictions? (b) If each couple is to sit together? (c) If all the men sit together to the right of all the women? Ans. 40, 320, 48, 576 12. Find the number of ways ways that 6 teachers can be assigned assigned to 4 sections of an introductory introductory psychology course if no teacher is assigned to more than one section. Ans. 360 PROBABILITY OF AN EVENT Perhaps it was man’s unquenchable thirst for gambling that led to the early development of probability theory. In an effort to increase their winnings, gamblers called upon mathematicians mathematicians to provide optimum strategies for various games of chance. Some of the mathematicians providing these strategies were Pascal, Leibniz, Fermat, and James Bernoulli . As a result of this early development of probability theory, statistical inference, with all its predictions and generalizations, has branched out far beyond games of chance to encompass many other fields associated with chance occurrences, such as politics, business, weather forecasting, and scientific research.
The modern period of probability theory is connected with names like S.N. Bernstein (1880-1968), E. Borel (1871-1956), and A.N. Kolmogorov (1903-1987). In particular, in 1933 A.N. Kolmogorov published his modern approach of probability theory, including the notion of a measurable space and probability space. 6
RB Astillero
Probability Theory
Probability of Simple Events
To study probability theory it is very important to have an understanding of its basic facts. What do we mean when we make the statements statements “John will probably win the tennis match,” or “I have a 50-50 50-50 chance of getting an even number when a die is tossed”? In each case we are expressing an outcome of which we are not certain, but owing to past information or from an understanding of the structure of the experiment, we have some degree of confidence in the validity of the statement. The likelihood of the occurrence of an event resulting from such a statistical experiment is evaluated by means of a set of real numbers called weights or probabilities ranging from 0 to 1. To every point in the sample space we assign a probability such that the sum of all probabilities is 1. If a certain sample point is quite likely to occur when the experiment is conducted, the probability assigned should be close to 1. On the other hand, a probability closer to 0 is assigned to a sample point that is not likely to occur. Definition The probability of an event A, denoted by the symbol P(A), is the sum of the weights of all sample points in A. Therefore, 0 P(A) 1, P() = 0, P(S) = 1 Furthermore, If A1, A2, A3,… is a sequence of mutually exclusive events then P(A1 A2 A3 …) = P(A1) + P(A2) + P(A3) + … Examples 1. A coin is tossed twice. What is the probability that at l east one head occurs? Solution: S = {HH, HT, TH, TT} Assuming the coin is balanced, each of the sample points would be equally likely to occur. Let w = weight of each sample point 4w = 1, w = ¼ A : event that at least 1 head occurs A = {HH, HT, TH} P(A) = ¼ + ¼ + ¼ = ¾ (ans) 2. A die is loaded in such a way that an even number is twice as likely to occur as an odd integer. If A is the event that a number less than 4 occurs on a single toss of a die, B is the event that an even number turns up, and C is the event that a number divisible by 3 occurs. Find (a) P(A), (b) P(B C), and (c) P(B C). Solution: S = {1, 2, 3, 4, 5, 6} A = {1, 2, 3} B = {2, 4, 6} C = {3, 6} Let w = weight of an odd integer 2w = weight of an even integer P(S) = w + 2w + w + 2w + w + 2w = 1 9w = 1 7
RB Astillero
Probability Theory
Probability of Simple Events
w = 1/9 P(A) = 1/9 + 2/9 + 1/9 = 4/9 (ans) P(B C) = 2/9 + 1/9 + 2/9 + 2/9 = 7/9 (ans) P(B C) = 2/9 (ans) Theorem 3.1 If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, then the probability of event A i s P( A)
=
The proof is left as an exercise Examples 1. A number between 1 and 10,000 inclusive is selected at random. What is the probability that it is a perfect square? Solution: A : the number selected is a perfect square S = {1, 2, 3,…,10,000} 2 2 2 2 A = {1 , 2 , 3 ,… ,100 } P(A) = n/N = 100/10,000 = 1/100 (ans) 2. In a poker hand consisting of 5 cards, find the probability of holding 2 aces and 3 jacks? Solution: A: the event of holding 2 aces B: the event of holding 3 jacks C: the event of holding 2 aces and 3 jacks n(A) = 4C2 = 4!/(2! 2!) = 6 n(B) = 4C3 = 4!/(3! 1!) = 4 n = 6 x 4 = 24 N = 52C5 = 52!/(5! 47!) = 2, 598, 960 -5 P(C) = n/N= 24/2, 598, 960 = 0.9 x 10 (ans) Exercises 3.3 1. The letters of the words PROBABILITY THEORY are written on identical plastic disks, which are then placed in a bowl and thoroughly mixed. If one of the disks is drawn at random, what is the probability that it has the letter B on it? Ans. 2/17 2. A box contains 10 white balls and 5 black balls. If 4 balls are drawn at random what is the probability that (a) all are black?, (b) 2 are white and 2 black?, (c) 3 are white and 1 black? Ans. 1/273, 90/273, 40/91
8
RB Astillero
Probability Theory
Probability of Simple Events
3. What is the probability that of 5 cards dealt from a well shuffled deck 3 will be hearts and 2 spades? Ans. 143/16, 660 4. Each of the numbers from 1 to 15 inclusive is written on a separate ticket. These tickets, which are identical in appearance, are thoroughly mixed and 2 of them are drawn at random. Find the probability that their sum will be 13. Ans. 2/35 5. Five red books and 3 blue books are placed at random on a shelf. Find the probability that the blue books will all be together. Ans. 3/28 6. Five red books, 4 blue books, and 3 green books are placed at random on a shelf. What is the probability that the blue books will all be together and the green books all together? Ans. 1/660 7. Certain manufactured articles are sold in boxes of 25 each. They are inspected by taking a random sample of 5 from each box and passing the box as satisfactory if no defective articles are found in the sample. Find the probability that a box will be passed if it contains (a) 1 defective, (b) 2 defectives, (c) 3 defectives. Ans. 4/5, 19/30, 57/115 8. Two cards are drawn in succession from a deck without replacement. What is the probability that both cards are greater than 2 and less than 8? Ans. 95/663 9. If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary, what is the probability that (a) the dictionary is selected? (b) 2 novels and 1 book of poems are selected? Ans. 1/3, 5/14 10. If 8 persons are seated at random in a row what is the probability that a certain couple (a) will be together? (b) will not be together? Ans. ¼, ¾ 11. If k persons are seated at random in a row of n seats show that the probability P that they will occupy consecutive seats is P = (n-k +1)! k!/n! 12. A pair of dice is thrown. If it is known that one die shows a 4, what is the probability that (a) the other die shows a 5? (b) the total of both dice is greater than 7? Ans. 2/11, 5/11 13. A 5-sided die with sides numbered 1, 2, 3, 4, and 5 is constructed so that the 1 and 5 occur twice as often as the 2 and 4, which occur three times as often as the 3. What is the probability that a perfect square occurs when this die is tossed once? Ans. 9/19 14. Three men are seeking public office. Candidates A and B are given about the same chance of winning, but candidate C is given twice the chance of either A or B. (a) What is the probability that C wins? (b) What is the probability that A does not win? Ans. 1/2, ¾
9
RB Astillero
Probability Theory
Probability of Simple Events
ADDITIVE RULES The Additive Rule is used to calculate the probability of some event from known probabilities of other events.
Theorem 3.2. Additive Rule If A and B are any two events, then P(A B) = P(A) + P(B) – P(B) – P(A B) The proof is left as an exercise [Hint: Use Venn diagram.] Corollary 1 If A and B are mutually exclusive, then P(A B) = P(A) + P(B) The proof is left as an exercise. Corollary 2 If A1, A2,…, An are mutually exclusive, then P(A1 A2 … An) = P(A1) + P(A2) + … + P(An) The proof is left as an exercise. Corollary 3 If A1, A2, … , An is a partition of a sample space S, then P(A1 A2 … An) = P(A1) + P(A2) + … + P(An) = P(S) =1 Theorem 3.3 For three events A, B, and C P(A B C) = P(A) + P(B) + P(C) – P(C) – P(A B) – B) – P(A C) – C) – P(B C) + P(A B C) The proof is left as an exercise. [Hint: Use Venn diagram] Theorem 3.4 If A and A’ are complementary events, then P(A) + P(A’) = 1 The proof is left as an exercise. Examples 1. A box contains 15 white balls, 4 black balls, and 6 red balls. What is the probability of drawing a black ball or a red ball in a single draw? SOLUTION: A: the ball is black B: the ball is red P(A B) = P(A) + P(B) [A and B ar e mutually exclusives] P(A) = 4/25 P(B) = 6/25 P(A B) = 4/25 + 6/25 = 2/5 (ans)
2. John is going to graduate from an industrial engineering department in a university by the end of the semester. After being interviewed at two companies he likes, he assesses that 10
RB Astillero
Probability Theory
Probability of Simple Events
his probability of getting an offer from company A is 0.8, and the probability that he gets an offer from company B is 0.6. if, on the other hand, he believes that the probability that he will get offers from both companies is 0.5, what is the probability that he will get at least one offer from these two companies? SOLUTION: P(A B) = P(A) + P(B) – P(B) – P(A B) = 0.8 + 0.6 – 0.6 – 0.5 = 0.9 (ans) 3. If the probabilities are respectively, 0.09, 0.15, 0.21, and 0.23 that a person purchasing a new automobile will choose the color green, white, red, or blue, what is the probability that a given buyer will purchase a new automobile that comes in one of those colors? SOLUTION: Let G, W, R, and B be the events that the buyer selects, respectively, a green, white, red, or blue automobile. Since these four events are mutually exclusive, the probability is P(G W R B) = P(G) + P(W) + P(R) + P(B) = 0.09 + 0.15 + 0.21 + 0.23 = 0.68 (ans) Exercises 3.4 1. According to records in the office of a College Registrar, 4 percent of the students in a certain course fail to pass. What is the probability that out of a random group of 4 students of the course exactly 2 will fail? Ans. 0.008847 2. A pair of dice is tossed. Find the probability of getting (a) a total of 8, (b) at most a total of 5. Ans. 5/36, 5/18 3. Two dice are thrown simultaneously. Find the probability that the sum of the number of spots uppermost is 7 or 11. Ans. 2/9 4. A box contains 500 envelopes of which 75 contain P100 in cash. 150 contain P25, and 275 contain P10. An envelope may be purchased for P25. Find the probability that the first envelope purchased contains less than P100. Ans. 17/20 5. Suppose that in a senior college class of 500 students it is found that 210 smoke, 258 drink alcoholic beverages, 216 eat between meals, 122 smoke and drink alcoholic beverages, 83 eat between meals and drink alcoholic beverages, 97 smoke and eat between meals, and 52 engage in all three of these bad health practices. If member of this senior class is selected at random, find the probability that the student (a) smokes but not drink alcoholic beverages; (b) eats between meals and drinks alcoholic beverages but does not smoke; and (c) neither smokes nor eats between meals. Ans. 22/125, 31/500, 171/500 6. The probability that an American industry will locate in Munich is 0.7, the probability that it will locate in Brussels is 0.4, and the probability that it will locate in either Munich or Brussels or both is 0.8. What is the probability that the industry will locate (a) in both cities? (b) in neither city? 11
RB Astillero
Probability Theory
Probability of Simple Events
Ans. 0.3, 0.2 7. In high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the probability that (a) the student took mathematics or history; (b) the student did not take either of these subjects; (c) the student took history but not mathematics. Ans. 22/25, 3/25, 17/50 8. According to Consumer Digest (July/August 1996), the probable location of PC’s in the home are: Adult bedroom – 0.03 Office or den – den – 0.40 Child bedroom – 0.15 Other rooms – rooms – 0.28 Other bedroom – bedroom – 0.14 (a) What is the probability that a PC is in a bedroom? (b) What is the probability that it is not in a bedroom? (c) Suppose a household is selected at random from households with PC; in what room would you expect to find a PC? Ans. 0.32, 0.68, office or den 9. In a certain manufacturing process it is found that, on the average, 2 articles out of 100 are defective. What is the probability that a random sample of 5 articles will contain (a) no defective articles? (b) exactly 1 defective article? (c) exactly 2 defective articles? Ans. 0.9039, 0.09224, 0.003765 10. Ten identical tickets are numbered from 1 to 10 inclusive, placed in a bowl and thoroughly mixed. A man draws a ticket at random. If it is numbered 5 or less he gets another draw, otherwise not. Find the probability that he draws a total of 8 or more (a) if the first ticket drawn is not replaced, (b) if it is replaced. Ans. 31/50, 3/5 11. The probability for the LET examinees from a certain school to pass the professional subjects is 4/9 and the major subjects is 7/9. If none of the examinees fails both subject and there are 30 examinees who pass both subjects, find the number of examinees from that school who took the LET. Ans. 135 12. In a certain federal prison, it is known that 2/3 of the inmates are under 25 years of age. It is also known that 3/5 of the inmates are male and the 5/8 of the inmates are female or 25 years of age or older. What is the probability that a prisoner selected at random from this prison is female and at least 25 years old? Ans. 13/120
12
