15 Chances, Probabilities, and Odds


 Conrad Rich
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1 15 Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces 15.2 Counting Outcomes in Sample Spaces 15.3 Permutations and Combinations 15.4 Probability Spaces 15.5 Equiprobable Spaces 15.6 Odds Excursions in Modern Mathematics, 7e:
2 Random Experiment Probability is the quantification of uncertainty. We will use the term random experiment to describe an activity or a process whose outcome cannot be predicted ahead of time. Examples of random experiments: tossing a coin, rolling a pair of dice, drawing cards out of a deck of cards, predicting the result of a football game, and forecasting the path of a hurricane. Excursions in Modern Mathematics, 7e:
3 Sample Space Associated with every random experiment is the set of all of its possible outcomes, called the sample space of the experiment. For the sake of simplicity, we will concentrate on experiments for which there is only a finite set of outcomes, although experiments with infinitely many outcomes are both possible and important. Excursions in Modern Mathematics, 7e:
4 Sample Space  Set Notation We use the letter S to denote a sample space and the letter N to denote the size of the sample space S (i.e., the number of outcomes in S). Excursions in Modern Mathematics, 7e:
5 Example 15.1 Tossing a Coin One simple random experiment is to toss a quarter and observe whether it lands heads or tails. The sample space can be described by S = {H, T}, where H stands for Heads and T for Tails. Here N = 2. Excursions in Modern Mathematics, 7e:
6 Example 15.2 More Coin Tossing Suppose we toss a coin twice and record the outcome of each toss (H or T) in the order it happens. What is the sample space? Excursions in Modern Mathematics, 7e:
7 Example 15.2 More Coin Tossing The sample space now is S = {HH, HT, TH, TT}, where HT means that the first toss came up H and the second toss came up T, which is a different outcome from TH (first toss T and second toss H). In this sample space N = 4. Excursions in Modern Mathematics, 7e:
8 Example 15.2 More Coin Tossing Suppose now we toss two distinguishable coins (say, a nickel and a quarter) at the same time. What is the sample space? Excursions in Modern Mathematics, 7e:
9 Example 15.2 More Coin Tossing The sample space is still S = {HH, HT, TH, TT}. (Here we must agree what the order of the symbols is for example, the first symbol describes the quarter and the second the nickel.) Excursions in Modern Mathematics, 7e:
10 Example 15.2 More Coin Tossing Since they have the same sample space, we will consider the two previous random experiments as the same random experiment. Excursions in Modern Mathematics, 7e:
11 Example 15.2 More Coin Tossing Suppose we toss a coin twice, but we only care now about the number of heads that come up. What is the sample space? Excursions in Modern Mathematics, 7e:
12 Example 15.2 More Coin Tossing Here there are only three possible outcomes (no heads, one head, or both heads), and symbolically we might describe this sample space as S = {0, 1, 2}. Excursions in Modern Mathematics, 7e:
13 Example 15.5 Dice Rolling The experiment is to roll a pair of dice. What is the sample space? Excursions in Modern Mathematics, 7e:
14 Example 15.5 More Dice Rolling Excursions in Modern Mathematics, 7e:
15 Example 15.5 More Dice Rolling Here we have a sample space with 36 different outcomes. Notice that the dice are colored white and red, a symbolic way to emphasize the fact that we are treating the dice as distinguishable objects. That is why the following rolls are distinguishable. Excursions in Modern Mathematics, 7e:
16 Example 15.5 More Dice Rolling The sample space has 36 possible outcomes: {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)} where the pairs represent the numbers rolled on each dice (white, red). Excursions in Modern Mathematics, 7e:
17 Example 15.4 Rolling a Pair of Dice Roll a pair of dice and consider the total of the two numbers rolled. What is the sample space? Excursions in Modern Mathematics, 7e:
18 Example 15.4 Rolling a Pair of Dice The possible outcomes in this scenario range from rolling a two to rolling a twelve, and the sample space can be described by S = {2, 3, 4, 5, 6, 7, 8, 9, 10,11,12}. Excursions in Modern Mathematics, 7e:
19 Examples Page 577, problem 3 Excursions in Modern Mathematics, 7e:
20 Page 577, problem 3 Solution: Examples {ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCAD, BCDA, BDAC, BDCA, CABD, CADB, CBAD, CBDA, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB, DCBA} There are 24 outcomes. Excursions in Modern Mathematics, 7e:
21 Not Listing All of the Outcomes We would like to understand what the sample space looks like without necessarily writing all the outcomes down. Our real goal is to find N, the size of the sample space. If we can do it without having to list all the outcomes, then so much the better. Excursions in Modern Mathematics, 7e:
22 15 Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces 15.2 Counting Outcomes in Sample Spaces 15.3 Permutations and Combinations 15.4 Probability Spaces 15.5 Equiprobable Spaces 15.6 Odds Excursions in Modern Mathematics, 7e:
23 Example 15.7 Tossing More Coins If we toss a coin three times and separately record the outcome of each toss, the sample space is given by S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Here we can just count the outcomes and get N = 8. Excursions in Modern Mathematics, 7e:
24 Example 15.7 Tossing More Coins Toss a coin 10 times In this case the sample space S is too big to write down We can count the number of outcomes in S without having to tally them one by one using the multiplication rule. Excursions in Modern Mathematics, 7e:
25 Multiplication Rule Suppose an activity consists of a series of events in which there are a possible outcomes for the first event, b possible outcomes for the second event, c possible outcomes for the third event, and so on. Then the total number of different possible outcomes for the series of events is: a b c Excursions in Modern Mathematics, 7e:
26 Example 15.7 Tossing More Coins Toss a coin ten times. How many outcomes are in the sample space? There are two outcomes on the first toss There are two outcomes on the second toss, etc. The total number of possible outcomes is found by multiplying ten two s together. Excursions in Modern Mathematics, 7e:
27 Example 15.7 Tossing More Coins Total number of outcomes if a coin is tossed ten times: N factors Thus N = 2 10 = Excursions in Modern Mathematics, 7e:
28 Example 15.8 The Making of a Wardrobe Dolores is a young saleswoman planning her next business trip. She is thinking about packing three different pairs of shoes, four skirts, six blouses, and two jackets. How many different outfits will she be able to create by combining these items? (Assume that an outfit consists of one pair of shoes, one skirt, one blouse, and one jacket.) Excursions in Modern Mathematics, 7e:
29 Example 15.8 The Making of a Wardrobe Let s assume that an outfit consists of one pair of shoes, one skirt, one blouse, and one jacket. Then to make an outfit Dolores must choose a pair of shoes (three choices), a skirt (four choices), a blouse (six choices), and a jacket (two choices). By the multiplication rule the total number of possible outfits is = 144. Excursions in Modern Mathematics, 7e:
30 Example Ranking the Candidate in an Election: Part 2 Five candidates are running in an election, with the top three vote getters elected (in order) as President, Vice President, and Secretary. How many different ways are there to choose the five candidates to fill these three positions? Excursions in Modern Mathematics, 7e:
31 Example Ranking the Candidate in an Election: Part 2 Excursions in Modern Mathematics, 7e:
32 Examples Page 578, problem 14 Excursions in Modern Mathematics, 7e:
33 Solution to part (a) Examples ,320 Excursions in Modern Mathematics, 7e:
34 Solution to part (a) Examples ,160 Excursions in Modern Mathematics, 7e:
35 Solution to part (c) Examples Excursions in Modern Mathematics, 7e:
36 15 Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces 15.2 Counting Outcomes in Sample Spaces 15.3 Permutations and Combinations (OMIT) 15.4 Probability Spaces 15.5 Equiprobable Spaces 15.6 Odds Excursions in Modern Mathematics, 7e:
37 Events An event is any subset of the sample space. That is, an event is any set of individual outcomes. This definition includes the possibility of an event that has no outcomes as well as events consisting of a single outcome. We denote events as E and the outcomes that make up E are listed inside braces { and } (that is, set notation) Excursions in Modern Mathematics, 7e:
38 Events An event that consists of no outcomes is an impossible event. For the impossible event we use the empty set so that E={ }. An event that consists of just one outcome is a simple event. Excursions in Modern Mathematics, 7e:
39 Example CoinTossing Event Suppose we toss a coin three times. The sample space for this experiment is: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }. Here N=8. Table 152 shows examples of events. Excursions in Modern Mathematics, 7e:
40 Example CoinTossing Event Excursions in Modern Mathematics, 7e:
41 Dice Rolling Recall the example The experiment is to roll a pair of dice, one red and one white. The sample space is depicted on the next page. Excursions in Modern Mathematics, 7e:
42 Example 15.5 More Dice Rolling Excursions in Modern Mathematics, 7e:
43 Events a) Write the event E of rolling a sum of 7. b) Write the event E of rolling the same numbers on both dice. c) Write the event of rolling at least one even number. Excursions in Modern Mathematics, 7e:
44 Events (a) Or E={(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} Excursions in Modern Mathematics, 7e:
45 Events (b) E={(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} Excursions in Modern Mathematics, 7e:
46 Events (c) E = {(1,2), (1,4), (1,6), (2,2), (2,4), (2,6), (3,2), (3,4), (3,6), (4,2), (4,4), (4,6), (5,2), (5,4), (5,6), (6,2), (6,4), (6,6),} Excursions in Modern Mathematics, 7e:
47 Examples Page 580, problem 44 Excursions in Modern Mathematics, 7e:
48 Solution to part (a) Examples E1 { TTFF, TFTF, TFFT, FTTF, FTFT, FFTT} Excursions in Modern Mathematics, 7e:
49 Solution to part (b) Examples E 2 { TTFF, TFTF, TFFT, FTTF, FTFT, FFTT, TTTF, TTFT, TFTT, FTTT, TTTT } Excursions in Modern Mathematics, 7e:
50 Solution to part (c) Examples E 3 { TTFF, TFTF, TFFT, FTTF, FTFT, FFTT, FFFT, FFTF, FTFF, TFFF, FFFF} Excursions in Modern Mathematics, 7e:
51 Solution to part (d) Examples E4 { TTFF, TTFT, TTTF, TTTT } Excursions in Modern Mathematics, 7e:
52 Probability Defined as the longterm proportion of times the outcome occurs Building Blocks of Probability Experiment  any activity for which the outcome is uncertain Outcome  the result of a single performance of an experiment Sample space (S)  collection of all possible outcomes Event  collection of outcomes from the sample space Excursions in Modern Mathematics, 7e:
53 Probability The probability for any event E is always between 0 and 1. If the event is impossible, its probability is 0. If the event is equal to the sample space, its probability is 1. Excursions in Modern Mathematics, 7e:
54 A probability assignment assigns to each simple event E in the sample space a number between 0 and 1, which represents the probability of the event E and which we denote by Pr(E). Pr({ })=0 Pr(S)=1 Probability Assignment Excursions in Modern Mathematics, 7e:
55 Probability Assignment Every probability assignment must obey the Law of Total Probability: For any experiment, the sum of all the outcome probabilities in the sample space must equal 1. Excursions in Modern Mathematics, 7e:
56 Example Handicapping a Tennis Tournament There are six players playing in a tennis tournament: A (Russian, female), B (Croatian, male), C (Australian, male), D (Swiss, male), E (American, female), and F (American, female). To handicap the winner of the tournament we need a probability assignment on the sample space S = {A, B, C, D, E, F }. Excursions in Modern Mathematics, 7e:
57 Example Handicapping a Tennis Tournament With sporting events the probability assignment is subjective (it reflects an opinion), but a professional oddsmaker comes up with the following probability assignment: Event A B C D E F Probability Excursions in Modern Mathematics, 7e:
58 Example Handicapping a Tennis Tournament Each probability is between 0 and 1. The sum of all outcome probabilities is 1 Pr( A) Pr( B) Pr( C) Pr( D) 0.25 Pr( E) Pr( F) Excursions in Modern Mathematics, 7e:
59 Probability Space Once a specific probability assignment is made on a sample space, the combination of the sample space and the probability assignment is called a probability space. Example: S = {A, B, C, D, E, F } Event A B C D E F Probability Excursions in Modern Mathematics, 7e:
60 Examples Page 580, problem 38 Excursions in Modern Mathematics, 7e:
61 Solution part (a) Pr(o 1 )=0.15= Pr(o 4 ) Pr(o 2 )=0.35= Pr(o 3 ) Examples Excursions in Modern Mathematics, 7e:
62 Solution part (b) Pr(o 1 )=0.15 Pr(o 2 )=0.35 Pr(o 3 )=0.22 Pr(o 4 )=0.28 Examples Excursions in Modern Mathematics, 7e:
63 15 Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces 15.2 Counting Outcomes in Sample Spaces 15.3 Permutations and Combinations 15.4 Probability Spaces 15.5 Equiprobable Spaces 15.6 Odds Excursions in Modern Mathematics, 7e:
64 Mutually Exclusive Events Events that have no outcomes in common are called mutually exclusive events. Excursions in Modern Mathematics, 7e:
65 Mutually Exclusive Events Example of mutually exclusive events: E 1 event of rolling a sum of 7 on two different color dice E 1 {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} E 2 event of rolling the same number on two different color dice E 2 {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} Excursions in Modern Mathematics, 7e:
66 Mutually Exclusive Events Example of events which are not mutually exclusive: E 1 event E 1 of rolling a sum of 7 on two different {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} color dice E 2 event E 2 of rolling at least one 6 on two different {(1,6),(2,6),(3,6),(4,6),(5,6),(6,6), color dice (6,1),(6,2),(6,3),(6,4),(6,5)} Excursions in Modern Mathematics, 7e:
67 Mutually Exclusive Events If events A and B are mutually exclusive, then Pr(A or B) = Pr(A)+Pr(B) If events A, B, and C are mutually exclusive, then Pr(A or B or C) = Pr(A)+Pr(B)+Pr(C) Etc. Excursions in Modern Mathematics, 7e:
68 Example Handicapping a (from 15.4) There are six players playing in a tennis tournament and the events in the sample space are: A (Russian, female), B (Croatian, male), C (Australian, male), D (Swiss, male), E (American, female), and F (American, female) Tennis Tournament Excursions in Modern Mathematics, 7e:
69 Example Handicapping a Tennis Tournament The probability space for the tournament winner is: S = {A, B, C, D, E, F } Event A B C D E F Probability Use the probability space to find the probability that an American will win the tournament. Excursions in Modern Mathematics, 7e:
70 Example Handicapping a Tennis Tournament Pr(E or F) = Pr(E) + Pr(F) = = 0.31 Excursions in Modern Mathematics, 7e:
71 Example Handicapping a Tennis Tournament What is the probability that a male will win the tournament? Excursions in Modern Mathematics, 7e:
72 Example Handicapping a Tennis Tournament Pr(B or C or D) =Pr(B) + Pr(C) + Pr(D) = = 0.61 Excursions in Modern Mathematics, 7e:
73 Example Handicapping a Tennis Tournament What is the probability that an American male will win the tournament? Excursions in Modern Mathematics, 7e:
74 Example Handicapping a Tennis Tournament Pr({ }) = 0 (since this one is an impossible event there are no American males in the tournament) Excursions in Modern Mathematics, 7e:
75 Calculating Probabilities Next we determine how to calculate probabilities for random experiments. Excursions in Modern Mathematics, 7e:
76 What Does Honesty Mean? What does honesty mean when applied to coins, dice, or decks of cards? It essentially means that all individual outcomes in the sample space are equally probable. Thus, an honest coin is one in which H and T have the same probability of coming up, and an honest die is one in which each of the numbers 1 through 6 is equally likely to be rolled. Excursions in Modern Mathematics, 7e:
77 Equiprobable Spaces A probability space in which each simple event has an equal probability of occurring is called an equiprobable space. Not all probability spaces are equiprobable as the next example shows. Excursions in Modern Mathematics, 7e:
78 Example Rolling a Pair of Honest Dice Suppose that you are playing a game that involves rolling a pair of honest dice, and the only thing that matters is the total of the two numbers rolled. The sample space in this situation is S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, where the outcomes are the possible totals that could be rolled. Excursions in Modern Mathematics, 7e:
79 Example Rolling a Pair of Honest Dice This sample space would not represent an equiprobable space. For example, the likelihood of rolling a 7 is much higher than that of rolling a 12. Excursions in Modern Mathematics, 7e:
80 Equiprobable Spaces If we know the probability space is an equiprobable space, we can determine the probability of any outcome as follows. Excursions in Modern Mathematics, 7e:
81 PROBABILITIES IN EQUIPROBABLE SPACES If k denotes the size of an event E and N denotes the size of the sample space S, then in an equiprobable space Pr E k N Excursions in Modern Mathematics, 7e:
82 Example Honest Coin Tossing Suppose that a coin is tossed three times, and we have been assured that the coin is an honest coin. If this is true, then each of the eight possible outcomes in the sample space: {HHH HHT HTH THH TTH THT HTT TTT} has probability 1/8. Excursions in Modern Mathematics, 7e:
83 Example Honest Coin Tossing This table shows each of the events with their respective probabilities. Excursions in Modern Mathematics, 7e:
84 Example More Dice Rolling If we roll a pair of honest dice (one red and one white), each of the 36 outcomes is equally likely: S = {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)} where the pairs represent the numbers rolled on each dice (white, red). Excursions in Modern Mathematics, 7e:
85 Example Rolling a Pair of Honest Dice Because the dice are honest, each of these 36 possible outcomes is equally likely to occur, so the probability of each is 1/36. Excursions in Modern Mathematics, 7e:
86 Example Rolling a Pair of Honest Dice Table 155 (next slide) shows the probability of rolling a sum of 2, 3, 4,..., 12. Excursions in Modern Mathematics, 7e:
87 Excursions in Modern Mathematics, 7e:
88 Examples Page 581, problem 50 (a) and (c) Excursions in Modern Mathematics, 7e:
89 Example Rolling a Pair of Honest Dice There are 16 outcomes in this sample space: N Excursions in Modern Mathematics, 7e:
90 Solution to part (a) Examples E1 { TTFF, TFTF, TFFT, FTTF, FTFT, FFTT} which has k=6 outcomes so that Pr( E 1 ) Excursions in Modern Mathematics, 7e:
91 Solution to part (c) Examples E 3 { TTFF, TFTF, TFFT, FTTF, FTFT, FFTT, FFFT, FFTF, FTFF, TFFF, FFFF} which has k=11 outcomes so that Pr( E 3 ) Excursions in Modern Mathematics, 7e:
92 Example Find the probability of drawing an ace when drawing a single card at random from a standard deck of 52 cards. Excursions in Modern Mathematics, 7e:
93 Solution Example continued The sample space for the experiment: Excursions in Modern Mathematics, 7e:
94 Example continued If a card is chosen at random, then each card has the same chance of being drawn. There are 52 outcomes in this sample space, so N = 52. Excursions in Modern Mathematics, 7e:
95 Example 5.1 continued Let E be the event that an ace is drawn. Event E consists of the four aces {A, A, A, A }, so k = 4. Therefore, the probability of drawing an ace is Pr(E) = 4/52 = 1/13 = Excursions in Modern Mathematics, 7e:
96 Example Experiment: A pair of dice are rolled. Define the following events A event the sum of the two diceequals 5 B event the sum of the two dice is 2 Find the probability that the sum of the two dice equals five or the sum of the two dice equals 2. Excursions in Modern Mathematics, 7e:
97 Example The event that the sum of the two dice equals five or the event that the sum of the two dice equals two are mutually exclusive and Pr( A or B) Pr( A) Pr( B) Excursions in Modern Mathematics, 7e:
98 Example The event that the sum of the two dice equals five. 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 A {(1,4),(2,3),(3,2),(4,1)} Excursions in Modern Mathematics, 7e:
99 Example The probability that the sum of the two dice equals five. Pr( A) k N Excursions in Modern Mathematics, 7e:
100 Example The event that the sum of the two dice equals two. 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 B {(1,1)} Excursions in Modern Mathematics, 7e:
101 Example The probability that the sum of the two dice equals two. Pr( B) k N Excursions in Modern Mathematics, 7e:
102 Example Pr( A or B) Pr( A) Pr( B) Excursions in Modern Mathematics, 7e:
103 Example Page 581, problem 55 (d) Excursions in Modern Mathematics, 7e:
104 Example The outcomes in the sample space consist of a sequence 10 of correct and incorrect guesses (possibly all correct and possibly all incorrect. For example, if we denote the event of getting an answer correct as C and incorrect as I we have outcomes like: {ICICICICCI, CIIICCICCC, ETC.} Excursions in Modern Mathematics, 7e:
105 Example Each sequence has an associated score: {ICICICICCI, CIIICCICCC, ETC.} score score Excursions in Modern Mathematics, 7e:
106 Example There are 1024 outcomes in this sample space: N Excursions in Modern Mathematics, 7e:
107 Example To get 8 or more points you can get all 10 correct (score=10 points) or 9 correct and 1 incorrect (score=8.5 points) Denote: A = event get all ten correct B = event you get 9 correct and 1 incorrect Excursions in Modern Mathematics, 7e:
108 Example There is one outcome in the sample space that corresponds to getting all ten correct: A={CCCCCCCCCC} Pr(A) Excursions in Modern Mathematics, 7e:
109 Example There are 10 outcomes in the sample space to getting 9 correct (C) and 1 incorrect (I): B={CCCCCCCCCI, CCCCCCCCIC, ETC.} Pr(B) Excursions in Modern Mathematics, 7e:
110 Example Rolling a Pair of Honest Dice Since events A and B are mutually exclusive we get that Pr( A or B) Pr( A) Pr( B) Excursions in Modern Mathematics, 7e:
111 Complement of E The complement of an event E is denoted: E C Collection of outcomes in the sample space that are not in event E Complement comes from the word to complete Any event and its complement together make up the complete sample space Excursions in Modern Mathematics, 7e:
112 Example If a fair coin is tossed three times. E = event that exactly one heads occurs (for example, HTT) What is the complement of E = E C? Excursions in Modern Mathematics, 7e:
113 Sample space: Example S { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT } Event E: E { HTT, THT, TTH} Complement of E: E C { HHH, HHT, HTH, THH, TTT } Excursions in Modern Mathematics, 7e:
114 Example: find the probability of the complement of an event E is the event observing a sum of 4 when the two fair (honest) dice are rolled, Find the probability that you do not roll a 4. Pr(E C ) Excursions in Modern Mathematics, 7e:
115 Example continued Solution Which outcomes belong to E C? There are the following outcomes in E: {(3,1)(2,2)(1,3)}. Excursions in Modern Mathematics, 7e:
116 Example continued All the outcomes except the outcomes from A in the twodice sample space: Excursions in Modern Mathematics, 7e:
117 Example continued There are 33 outcomes in E C and 36 outcomes in the sample space. This gives the probability of not rolling a 4 to be Pr(E C ) = 33/36 = 11/12 = The probability is high that, on this roll at least, your roommate will not land on Boardwalk. Excursions in Modern Mathematics, 7e:
118 Probabilities for Complements For any event E and its complement E C, Pr(E) + Pr(E C ) = 1 OR: Pr(E) = 1  Pr(E C ) OR: Pr(E C ) = 1  Pr(E) Excursions in Modern Mathematics, 7e:
119 Example Consider the experiment of drawing a card at random from a shuffled deck of 52 cards. Find the probability of drawing a card that is not a face card. NOTE: a face card is a king, queen, or jack Excursions in Modern Mathematics, 7e:
120 Example The sample space for the experiment where a subject chooses a single card at random from a deck of cards is depicted here. Excursions in Modern Mathematics, 7e:
121 ANSWER: Example There are 52 cards, 12 of which are face cards. E event you draw a facecard C E Pr( E) k N event you do not draw a facecard Pr( E C ) Excursions in Modern Mathematics, 7e:
122 Example Handicapping a Tennis Tournament There are six players playing in a tennis tournament and the events in the sample space are: A (Russian, female), B (Croatian, male), C (Australian, male), D (Swiss, male), E (American, female), and F (American, female) Excursions in Modern Mathematics, 7e:
123 Example Handicapping a Tennis Tournament The probability space for the tournament winner is: S = {A, B, C, D, E, F } Event A B C D E F Probability Use the probability space to find the probability that a Russian will not win the tournament. Excursions in Modern Mathematics, 7e:
124 Example Handicapping a Tennis Tournament The probability a Russian will win the tournament is: Pr(A) 0.08 The probability a Russian will not win the tournament is: Pr( A C ) Excursions in Modern Mathematics, 7e:
125 Independent Events Two events are said to be independent events if the occurrence of one event does not affect the probability of the occurrence of the other. Excursions in Modern Mathematics, 7e:
126 Independent Events Examples of independent events: 1.Flip a coin three times; the outcome of any one flip does not affect the probability of another flip. 2.Roll a die four times; the outcome of any one roll does not affect the probability of another roll. Excursions in Modern Mathematics, 7e:
127 Example Rolling a Pair of Honest Dice Imagine a game in which you roll an honest die four times. Find the probability that you roll a number that is not one on all four rolls of the dice. Excursions in Modern Mathematics, 7e:
128 Example Rolling a Pair of Honest Dice There are 1296 outcomes in this sample space: N k = number of events corresponding to the event that you roll a number that is not one on all four rolls of the dice Excursions in Modern Mathematics, 7e:
129 Example Rolling a Pair of Honest Dice: Part 3 Let R 1, R 2, R 3, and R 4 denote the outcomes on roll 1, roll 2, roll 3, and roll 4. Examples of outcomes that you do not roll 1: R 1 = 5, R 2 = 3, R 3 = 4, R 4 = 2 R 1 = 4, R 2 = 6, R 3 = 2, R 4 = 2 It is not practical to try and list all of these events to find k. Excursions in Modern Mathematics, 7e:
130 Independent Events When events E and F are independent, the probability that both occur is the product of their respective probabilities; in other words, Pr(E and F) = Pr(E) Pr(F) This is called the multiplication principle for independent events. Excursions in Modern Mathematics, 7e:
131 Example Rolling a Pair of Honest Dice Let F 1, F 2, F 3, and F 4 denote the events first roll is not a one, second roll is not a one, third roll is not a one, and fourth roll is not a one, respectively. These events are independent. Let F = F 1 and F 2 and F 3 and F 4 Excursions in Modern Mathematics, 7e:
132 Example Rolling a Pair of Honest Dice Then Pr(F 1 ) = 5/6, Pr(F 2 ) = 5/6 Pr(F 3 ) = 5/6, Pr(F 4 ) = 5/6 Multiplication principle for independent events gives the probability that you roll a number that is not one on all four rolls of the dice. Pr(F) = 5/6 5/6 5/6 5/6 = (5/6) Excursions in Modern Mathematics, 7e:
133 Example Rolling a Pair of Honest Dice Imagine a game in which you roll an honest die four times. If at least one of your rolls comes up a one, you are a winner. Let E denote the event you win. Find Pr(E). Excursions in Modern Mathematics, 7e:
134 Example Rolling a Pair of Honest Dice Let R 1, R 2, R 3, and R 4 denote the outcomes on roll 1, roll 2, roll 3, and roll 4. Examples of rolls where you are a winner: R 1 = 4, R 2 = 3, R 3 = 1, R 4 = 6 R 1 = 1, R 2 = 5, R 3 = 2, R 4 = 1 It would be tedious to write out all possible events in the sample space that correspond to at least one of the rolls comes up a one. Excursions in Modern Mathematics, 7e:
135 Example Rolling a Pair of Honest Dice The complement of E is the event that none of the rolls comes up one; that is, the event that you roll a number that is not one on all four rolls of the dice. Therefore we use the previous example and the complement rule for probabilities to find the answer. Excursions in Modern Mathematics, 7e:
136 Example Rolling a Pair of Honest Dice Pr(E) = 1 Pr(F) = = Excursions in Modern Mathematics, 7e:
137 15 Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces 15.2 Counting Outcomes in Sample Spaces 15.3 Permutations and Combinations 15.4 Probability Spaces 15.5 Equiprobable Spaces 15.6 Odds Excursions in Modern Mathematics, 7e:
138 Example Odds of Making In Example discusses the fact that Steve Nash (one of the most accurate freethrow shooters in NBA history) shoots free throws with a probability of p = We can interpret this to mean that on the average, out of every 100 free throws,nash is going to make 90 and miss about 10, for a hit/miss ratio of 9/1 or 9 to 1. Free Throws Excursions in Modern Mathematics, 7e:
139 Example Odds of Making Free Throws This ratio of hits to misses gives what is known as the odds of the event (in this case Nash making the free throw). Excursions in Modern Mathematics, 7e:
140 ODDS Let E be an arbitrary event. If F denotes the number of ways that event E can occur (the favorable outcomes or hits) and U denotes the number of ways that event E does not occur (the unfavorable outcomes, or misses), then the odds of (also called the odds in favor of) the event E are given by the ratio: F/U or F to U Excursions in Modern Mathematics, 7e:
141 ODDS the odds against the event E are given by the ratio U/F or U to F Note: the odds of E is the same as the odds in favor of E Excursions in Modern Mathematics, 7e:
142 Example Odds of Rolling a Natural Suppose that you are playing a game in which you roll a pair of dice, presumably honest. In this game, when you roll a natural (i.e., roll a sum of 7 or sum of 11) you automatically win. Find the odds of winning. Excursions in Modern Mathematics, 7e:
143 Example Odds of Rolling a Natural If we let E denote the event roll a natural, we can check that out of 36 possible outcomes 8 are favorable: 6 ways to roll a 7 plus two ways to roll an 11 and the other 28 are unfavorable. It follows that the odds of rolling a natural are 8/28=2/7 or 2 to 7 Excursions in Modern Mathematics, 7e:
144 Converting Odds to Probability To convert odds into probabilities: If the odds of E are F/U or F to U, then Pr(E) = F/(F + U) Excursions in Modern Mathematics, 7e:
145 Example Page 582, problem 61 Excursions in Modern Mathematics, 7e:
146 Example Page 582, problem 61 (a)3/8 (b)15/23 Excursions in Modern Mathematics, 7e:
147 Converting Probability to Odds To convert probabilities into odds when the probability is given in the form of a fraction: If Pr(E) = A/B then the odds of E are: B A A or (When the probability is given in decimal form, the best thing to do is to first convert the decimal form into fractional form.) A to B A Excursions in Modern Mathematics, 7e:
148 Example Page 582, problem 60 Excursions in Modern Mathematics, 7e:
149 Example Page 582, problem 60 (a)3/8 (b)3/5 Excursions in Modern Mathematics, 7e:
150 Casinos and Bookmakers Odds There is a difference between odds as discussed in this section and the payoff odds posted by casinos or bookmakers in sports gambling situations. Suppose we read in the newspaper, for example, that the Las Vegas bookmakers have established that the odds that the Boston Celtics will win the NBA championship are 5 to 2. Excursions in Modern Mathematics, 7e:
151 Casinos and Bookmakers Odds What this means is that if you want to bet in favor of the Celtics, for every $2 that you bet, you can win $5 if the Celtics win. The ratio 5 to 2 may be taken as some indication of the actual odds in favor of the Celtics winning, but several other factors affect payoff odds, and the connection between payoff odds and actual odds is tenuous at best. Excursions in Modern Mathematics, 7e:
152 Casinos and Bookmakers Odds This ratio may be taken as some indication of the actual odds in favor of the Celtics winning, but several other factors affect payoff odds, and the connection between payoff odds and actual odds is tenuous at best. Excursions in Modern Mathematics, 7e:
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