CHAPTER 7 Probability

Transcription

1 CHAPTER 7 Probability 7.1. Sets A set is a well-defined collection of distinct objects. Welldefined means that we can determine whether an object is an element of a set or not. Distinct means that we can tell the objects apart. Basic Set Definitions Well-defined Set: The set of natural numbers between 3 and 7, inclusive: {1, 2, 3, 4, 5, 6, 7}. The set of rational numbers: { a b a, b are integers, b 0 }. Not Well-defined Set: The set of large numbers. Set with Non-distinct Objects: {A, B, C, C} 29

2 30 HELENE PAYNE, FINITE MATHEMATICS Universal set - U: In a given problem, the single larger set from which all the elements of the sets under discussion are drawn. Empty set or Null set - or {}: A set containing no elements. Venn diagram: A tool used in visualizing sets invented by John Venn ( ). U A B Set definitions Definition Example is an element of 2 {0, 1, 2, 3, 4, 5, 6} / is not an element of 2 / {1, 3, 5} is a subset of {1, 3, 5} {0, 1, 2, 3, 4, 5, 6, 7, 8} {1, 3, 5} {1, 3, 5} is not a subset of {0, 1, 2, 3, 4, 5, 6} {1, 3, 5} is a proper subset of {3, 5} {1, 3, 5} is not a proper subset of {1, 3, 5} {1, 3, 5} = set equality {1, 3, 5} = {3, 1, 5} The order of elements does not matter

3 CHAPTER 7. PROBABILITY 31 Exercise 31. Determine whether A is a subset of B. Draw a Venn diagram for each case. (a) A = {2, 3, 4} and B = {1, 2, 3, 4, 5} (b) A = {0, 2, 3, 4} and B = {1, 2, 3, 4, 5}

4 32 HELENE PAYNE, FINITE MATHEMATICS The Complement and Set Definitions involving Two Sets For the examples in the table below, suppose set A = {0, 2, 4, 6, 8}, set B = {1, 3, 5, 7, 9}, set C = {1, 2, 3, 4} and the universal set, U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Set definitions Definition Example The intersection of two sets A B = A and B = the set of A C = {2, 4} all elements in both A and B The union of two sets A and B = A C = {0, 1, 2, 3, 4, 6, 8} the set of all elements A B = U in either A or B or both The complement of a set A, C = {0, 5, 6, 7, 8, 9} A = all elements in the Universal set, U, not in set A Two sets, A and B are disjoint, if A B =. A = {1, 3, 5, 7, 9} = B A and B are disjoint, since A B = B and C are not disjoint, since B C = {1, 3}

5 CHAPTER 7. PROBABILITY 33 Exercise 32. For the sets A = {a, b, c}, B = {c, e, f}, C = {g, h} and the universal set, U = {a, b, c, d, e, f, g, h, i, j}, find (a) A (b) B (c) A B (d) A B (e) (f) B C

6 34 HELENE PAYNE, FINITE MATHEMATICS Exercise 33. Let the universal set, U be the set of all Foothill College students, let E be the set of all Foothill College students enrolled in an English class, let G be the set of all Foothill College students enrolled in a geology class, and let M be the set of all Foothill College students enrolled in a math class. (a) Use the given set symbols to construct a Venn diagram identifying the various sets of students. (b) Use the given set symbols along with, or, to identify the set of students enrolled in English, but not in geology or math.

7 CHAPTER 7. PROBABILITY 35 Application of Venn Diagrams - Labeling Regions When working with sets using Venn diagrams, it is especially useful to think of the different regions of the Venn diagram as different states of a country, where the universal set would represent the entire country. We will give each state a different label (a letter or number). This will greatly simplify the process of shading regions in the Venn diagram corresponding to set expressions involving the symbols,, and. Exercise 34. By labeling regions in a Venn diagram, represent the following set: A B C. U A B C Set A B A B C A B C Region Labels

8 36 HELENE PAYNE, FINITE MATHEMATICS De Morgan s Properties Let A and B be sets, then 1. A B = A B 2. A B = A B Exercise 35. By labeling regions in a Venn diagram, show that: A B = A B. U A B Set Region Set Region Labels Labels A B A B A B A B A B

9 CHAPTER 7. PROBABILITY The Number of Elements in a Set n(a) denotes the number of elements in A, if A is a finite set. Exercise 36. Calculate n(a) for each set A below. (a) A = {1, 2, π, e, 11, 4 123}, n(a) = (b) A = the letters of the alphabet, n(a) = (c) A =, n(a) = Important Counting Formulas The Inclusion-Exclusion Principle 1. n(a B) = n(a) + n(b) n(a B), or n(a or B) = n(a) + n(b) n(a and B) The Complement Principle 2. n(a) = n(u) n(a) Exercise 37. Verify the counting formulas above using the Venn diagram below. Each element is represented by a diamond. U A B (a) n(a B) = n(a) = n(b) = n(a B) = (b) n(a) = n(u) = n(a) =

10 38 HELENE PAYNE, FINITE MATHEMATICS Exercise 38. A Chevrolet dealer has 160 new cars for sale on her lot. Among these cars, Fifty have four-cylinder engines. Eighty have a tilted steering wheel. Thirty have power windows. Forty-two have four-cylinder engines and a tilted steering wheel. Eighteen have four-cylinder engines and power windows. Fifteen have all three features. Sixty-five have none of the features. Let F be the set of cars with four-cylinder engines, T be the set of cars with a tilted steering wheel, and let P be the set of cars with power windows. Use the Venn diagram to answer the questions below. U A B C How many of these cars have (a) A tilted steering wheel and power windows?

11 CHAPTER 7. PROBABILITY 39 (b) At least one of these features? (c) Exactly two of these features Exercise 39. Suppose 250 domestic CEO s were surveyed about their company s industry type and geographic location in the United States. The CEO s were allowed to choose only one industry type (manufacturing, communications, or finance) and one location (Northeast, Southeast, Midwest, or West). The results are given below: Find North- South- Mideast east west West Manufacturing Communications Finance (a) The number of CEO s whose response was not Southeast. (b) The number of CEO s whose response was Communications or West. (c) The number of CEO s whose response was Northeast but not Manufacturing. (d) The number of CEO s whose response was Manufacturing or Communications or Midwest or West.

12 40 HELENE PAYNE, FINITE MATHEMATICS 7.3. The Multiplication Principle In this section, we will learn to count the number of elements in a set too big to list all elements. This is necessary to be able to calculate probabilities. a c b Exercise 40. A box contains three slips of paper with the letters a written on one, the letter b written on another and the letter c written on the third slip. A slip is drawn without replacement (=without putting it back), and the letter is recorded. Then a second slip is drawn and its letter is recorded. A typical outcome for this experiment is (b, a) if the first letter drawn was a b and the second one was a c. (a) List the set of outcomes, the sample space, S. How many different outcomes are there?

13 CHAPTER 7. PROBABILITY 41 (b) Make a tree diagram of this experiment Based on our findings in the tree diagram, suggest a formula for calculating the number of outcomes for this experiment. The outcomes in the previous experiment are called ordered arrangements since the order is important. (a, b) is a different outcome from (b, a). Number of outcomes:

14 42 HELENE PAYNE, FINITE MATHEMATICS Exercise 41. Repeat the previous experiment, but now, do it with replacement (=putting the slip back), after the letter is recorded. (a) List the set of outcomes, i.e. the sample space, S. How many different outcomes are there now? (b) Make a tree diagram of this experiment (c) Use the formula found in the previous exercise to calculate the number of outcomes when drawing two slips with replacement.

15 CHAPTER 7. PROBABILITY 43 The Multiplication Principle of Counting. An experiment is performed where there are p selections for the first choice, q selections for the second choice, r selections for the third choice and so on, p =the number of selections for the first choice q =the number of selections for the second choice r =the number of selections for the third choice. Then the number of distinct ordered outcomes, n(s) is: (7.1) n(s) = p q r Exercise 42. A coin is tossed 3 times and each time H (head) or T (tail) is recorded. A typical outcome for this experiment is HT T which represents the outcome where the first toss was a head and the second and third tosses were tails. (a) List the sample space for this experiment. (b) Use the multiplication principle to determine the total number of outcomes for this experiment. (c) Use the multiplication principle to determine how many outcomes have a head on the first toss?

16 44 HELENE PAYNE, FINITE MATHEMATICS Exercise 43. Tom is getting dressed for work. He will wear a shirt, a pair of pants and a jacket. He owns 4 shirts, 3 pairs of pants and 2 jackets. How many different outfits could he wear? Exercise 44. How many different license plates could be made using three letters followed by three digits if (a) there are no restrictions? (b) no letter can be repeated? (c) no letter can be repeated and zero cannot be used as the first digit? Exercise 45. In how many ways can five of nine people be seated in five chairs?

17 CHAPTER 7. PROBABILITY Sample Spaces and the Assignment of Probabilities. The foundations of modern probability theory was laid by the French mathematicians, Blaise Pascal ( ) and Pierre de Fermat ( ). The need for probability theory came from games of chance (=gambling) as well as mortality tables and insurance rates. Probability: The chance or likelihood of an event occurring. Experiment: An activity or procedure that produces distinct, well-defined possibilities called outcomes. Outcomes: Observations or measurements from a probability experiment, which cannot be predicted with certainty. Sample Space, S: The set of all possible outcomes, i.e. the universal set of outcomes. Event: An set of outcomes from a probability experiment. An event is a subset of S, the sample space. Simple Event: An event consisting of a single outcome in sample space. In this section, we will learn to assign a probability to various events in a probability experiment, by making a probability model. We will begin by studying models for which each outcome has the same probability.

18 46 HELENE PAYNE, FINITE MATHEMATICS Exercise 46. A balanced coin is to be tossed, and the result of each toss is recorded. Find the probability of each outcome in the sample space. Sample space, S = { Assign a probability for each outcome in this experiment. List the probability distribution in the table below: Outcome, e n Probability of Outcome, P (e n ) Exercise 47. Toss a coin 50 times and record number of heads and tails in the table below. Number Number Empirical Empirical of Heads of Tails Probability Probability of Heads of Tails As you can see, the empirical probabilities are not usually equal to 1 2, but close. When we say that the probability of getting a head is 1 2 we mean that in the long run if the experiment is repeated over and over, on average, we will get a head 1 2 the time.

19 CHAPTER 7. PROBABILITY 47 Exercise 48. A pair of dice, one red and one green is to be rolled and the number of pips on the upper face recorded. A typical outcome forthis experiment is (2, 3) meaning the red die showed a 2 and the green die showed a 3. Complete the table below listing the sample space for this experiment. Red Die Green Die (a) How many outcomes are there? n(s) = (b) What is the probability for each outcome in sample space? (c) Let E be the event: E: A sum of 5 is rolled. List the outcomes in E: E = { (d) What is the probability of rolling a sum of 5, i.e. find P (E):

20 48 HELENE PAYNE, FINITE MATHEMATICS Exercise 49. Complete the table below, listing the sum of the red and green dice: Red Die Green Die (a) What is the probability of rolling a sum of 7? (b) What is the probability of rolling a sum less than 7? Probability of an Outcome Suppose the sample space of an experiment has n outcomes given by: S = {e 1, e 2,..., e n } The following must hold for the probabilities of these events:

21 1. 0 P (e 1 ) 1, 0 P (e 2 ) 1,. 0 P (e n ) 1 CHAPTER 7. PROBABILITY 49 - The probability of each outcome must be a number between 0 and P (e 1 + P (e 2 ) P (e n ) = 1 - The sum of the probabilities of all outcomes of sample space must equal 1 The Probability of an Event Let S be a sample space with n = n(s) equally likely outcomes. The the probability of each outcome is (7.2) 1 n(s). If E is an event in this sample space with m = n(e) outcomes, then the probability of E, P (E) is (7.3) P (E) = n(e) n(s) = m n.

22 50 HELENE PAYNE, FINITE MATHEMATICS Exercise 50. Dr. Sun, a mathematics professor, gives a two-question, multiple-choice quiz in which each question has four possible answers: A, B, C, and D. Assume that the student guesses at the answers on this quiz. (a) Write the outcomes of the sample space for this experiment. (b) Assign a probability to each outcome. (c) What is the probability that the student will answer both questions correctly? (d) What is the probability that the student will answer one question correctly?

23 CHAPTER 7. PROBABILITY 51 Exercise 51. A single card is to be drawn from a standard deck of 52 cards. (a) How many outcomes are in the sample space for this experiment? n(s) = (b) What is the probability of drawing a club? (c) What is the probability of drawing a king? Exercise 52. A ball is picked at random from a box containing three red, five green balls, seven blue and four white balls. What is the probability that (a) a white ball is picked? (b) a green ball is picked? (c) a blue or red ball is picked?

24 52 HELENE PAYNE, FINITE MATHEMATICS Exercise 53. The number of new home sales (in thousands) in the United States as of March, 24, 2010, is given below. Northeast Midwest South West Feb-Mar Apr-Jun Jul-Sep Oct-Dec Jan-Feb (a) What is the probability a home selected at random was sold in the south? (b) What is the probability a home selected at random was sold from Apr-Jun 2009? (c) What is the probability a home selected at random was sold in the south from Apr-Jun 2009?

25 CHAPTER 7. PROBABILITY Properties of the Probability of an Event Event; Simple Event An event is any subset of a sample space. If an event has exactly one element, that is, if it consists of only one outcome, it is called a simple event. Exercise 54. Consider the experiment of selecting one family from the set of all possible three-child families. The simple event GGB is a family whose children are girl, girl, then boy, in that order. (a) List the sample space for this experiment: S = { (b) List the event that the family had a girl first, then two boys. E = { (c) List the event, F that the family had exactly one boy. F = { Notice that event F is not a simple event, but can be expressed as the union of the three simple events: {BGG}, {GBG}, and {GGB}: F = {BGG} {GBG} {GGB}

26 54 HELENE PAYNE, FINITE MATHEMATICS General Rules for Probability. The sum of the probabilities of the simple events of sample space, e n, is always 1, i.e., P (S) = P (e 1 ) + + P (e n ) = 1. The probability of any event, E, is always a number between 0 and 1, i.e. 0 P (E) 1. The probability of an empty event,, is always zero, i.e. P ( ) = 0. The probability of an event, E is found by adding the probabilities of all simple events in sample space making up E, i.e. P (E) = P (e 1 )+P (e 2 )+ +P (e r ). Exercise 55. Suppose a particular experiment results in the sample space S = {a, b, c, d, e}. (a) If all outcomes in S are equally likely, what is the probability of a single outcome? (b) If not all outcomes are equally likely, P (a) = 0.1, P (b) = 0.3, P (c) = 0.2, and P (d) = 0.1, what is P (e)? (c) If E = {a, b, d, e} is an event in S with probability 0.8, what is P (c)? (d) Explain why the probability assignment P (a) = 0.4, P (b) = 0.3, P (c) = 0.1, P (d) = 0.4, and P (e) = 0.2 is not valid.

27 CHAPTER 7. PROBABILITY 55 Mutually Exclusive Events. Two events, E and F are mutually exclusive if they cannot occur simultaneously (they share no outcomes), i.e. E F =. Probability of E or F for Mutually Exclusive Events. If two events, E and F are mutually exclusive, i.e. E F =, then the probability of E or F is the sum of their probabilities: P (E or F ) = P (E) + P (F ), or P (E F ) = P (E) + P (F ). Exercise 56. The following data were collected from a finite mathematics class at State University. Have a Scholarship Freshman 8 5 Sophomore 5 7 Junior 3 6 No Scholarship (a) What is the probability that the student chosen is a sophomore (b) What is the probability that the student chosen is a freshman or junior?

28 56 HELENE PAYNE, FINITE MATHEMATICS All events are not mutually exclusive. In the previous exercise, suppose we wanted to find the probability that a student is a sophomore or has a scholarship. The events, having a scholarship and being a sophomore are not mutually exclusive as a student could be a sophomore with a scholarship. We will use the Inclusion-Exclusion Principle to calculate probabilities. It is: n(a B) = n(a) + n(b) n(a B) The formula for calculating the probability of the union of two events is analogous to the inclusion-exclusion principle is listed next: Additive Rule for Probability: The Union of Two Events For any two events E and F in a sample space P (E F ) = P (E) + P (F ) P (E F ), or P (E or F ) = P (A) + P (B) P (A and B) Exercise 57. Referring back to the previous exercise with the data were collected from a finite mathematics class at State University. (a) What is the probability that the student chosen is a sophomore and has a scholarship?

29 CHAPTER 7. PROBABILITY 57 (b) What is the probability that the student chosen is a sophomore or has a scholarship? Exercise 58. Let S be the sample space in which the events A and B are such that P (A) = 0.4, P (B) = 0.5, and P (A B) = 0.2. (a) Illustrate these probabilities with a Venn diagram. (b) Use the Venn diagram to find the probability of the event A or B, i.e. A B, and compare the answer with P (A) + P (B). Are events A and B mutually exclusive? (c) Use the Venn diagram to find the probability of A but not B, i.e. P (A B).

30 58 HELENE PAYNE, FINITE MATHEMATICS Exercise 59. An auditor of personal income tax forms submitted to a certain state gathered the following information from a randomly selected sample of 300 forms: Income Level Correct Understated Tax $30, 000 $70, 000 $70, 000 $120, Overstated Tax (a) Find the probability that a person will submit a correct form? (b) Find the probability that a person will be in the $30, 000 $70, 000 income level and will submit a taxform that understates his or her tax? (c) Find the probability that a person will be in the $30, 000 $70, 000 income level or will submit a taxform that understates his or her tax?

31 CHAPTER 7. PROBABILITY 59 If we calculate the probability that an event will occur, how is that related to the probability of that event not occurring? In other words, how do we find the probability of the complement of an event? Since we must always have one or the other, either an event E will occur, or it won t (i.e. its complement occurs) and they are mutually exclusive, we have: P (E E) = P (E) + P (E) = 1. From this we derive the formula for the probability of the complement of an event E, P (E): Probability of the Complement of an Event (7.4) P (E) = 1 P (E) Exercise 60. For a certain weighted die, the probability of getting a six is 0.1. What is the probability of not getting a six? Exercise 61. What is the probability that a seven digit telephone number has one or more repeated digits? Note that the telephone number cannot begin with a zero.

32 60 HELENE PAYNE, FINITE MATHEMATICS Exercise 62. What is the probability that in a group of 20 people, at least 2 people have the same birthday? Another form of probability is in the statement of odds for or against some event. For example, in a local soccer league, the odds are 5 to 3 that the team Sting will beat the team Mirage. This means that if a series of 8 games (5+3 = 8) is played over and over, the Sting would tend to win 5 of the games, and Mirage 3 of the games. Therefore, the probability that Sting will win is 5 8 and the probability that Mirage will win is 3 8. In general, we have: Odds Formula. If the odds for an event E is stated as a to b, then probability that event E will occur, P (E) is: (7.5) P (E) = a a + b

33 and the probability of E, P (E) is: CHAPTER 7. PROBABILITY 61 (7.6) P (E ) = b a + b On the other hand, the odds for an event E, a to b are found by dividing P (E) by P (E ) and reducing to its lowest terms, a b : (7.7) The odds for event E, P (E) P (E) = a b, and similarly, (7.8) the odds against event E, P (E) P (E) = b a. Exercise 63. A fast-food chain is conducting a game in which the odds of winning a double cheeseburger are reported to be 1 : 100. Find the probability of winning a double cheeseburger.

34 62 HELENE PAYNE, FINITE MATHEMATICS Exercise 64. A pair of dice, one red and one green, is to be rolled. Find the odds (a) for the sum of the pips on the top sides to be 6. (b) for the sum of the pips on the top sides to be 11. (c) for the red die to show three pips on the top side. (d) against the pips sum on the top sides being 3.

35 7.6. Expected Value CHAPTER 7. PROBABILITY 63 Whenever a probability distribution has numerical outcomes, i.e. outcomes that are numbers. We can calculate average or expected value for the outcomes. Suppose you are playing the following card game. You draw a card at random from a deck of cards. If it is a face card, you win $2. If it is a non-face card, you lose $1. What are your expected earnings from this game? Outcome Payoff Assigned face card non-face card = Earnings Probability The average earnings after repeating this experiment 1000 times is: $ ( $1) = $2 3 +( $1) $0.31 This means that on average, a player will lose about 31 cents per draw. If we had calculated the average for 10, 000 draws, 100, 000 draws or any other number of draws, we would obtain the same value for the expected earnings.

36 64 HELENE PAYNE, FINITE MATHEMATICS Expected Value. Let S be a sample space, and let A 1, A 2,, A n be n events of S that form a partition of S, that is the union of the events is S and they are pairwise disjoint: A 1 A 2... A n = S and A i A j = for all i j. Let p 1, p 2, p 3,..., p n, be the probabilities of the events A 1, A 2,, A n. If each event, A 1, A 2,, A n is assigned the payoffs, m 1, m 2,, m n the expected value corresponding to these payoffs is: (7.9) E = m 1 p 1 + m 2 p m n p n. Exercise 65. To move from your current position on a game board, you draw one ball, with replacement, from a container that has five red and six green balls. If the ball you draw is red, you advance four places; if the ball is green, you move back three places. (a) What is your expected movement in this game? (b) In 44 turns at drawing a ball, where would you expect to be located on the game board relative to your present position?

37 CHAPTER 7. PROBABILITY 65 Exercise 66. Assume that a boy is just as likely as a girl at each birth. In a three-child family, what is the expected number of boys? Exercise 67. Suppose you pay $5 to play this game: From a box containing six red and eight green marbles, you are allowed to select one marble after being blindfolded. If the marble is red, you win $10, but if the marble is green, you win nothing. What are your expected earnings from this game?

38 66 HELENE PAYNE, FINITE MATHEMATICS Exercise 68. A department store wants to sell 11 purses that cost the store $40 each, and 32 purses that cost the store $10 each. If all purses are wrapped in 43 identical boxes, and if each customer picks a box randomly, find (a) Each customer s expectation (=average value of purse picked). (b) The department store s expected profit per purse, if it charges $20 for each box.