3 Chapter 5 Probability 2010 Pearson Prentice Hall
3 Chapter 5 Probability © 2010 Pearson Prentice Hall. All rights reserved
Section 5. 1 Probability Rules © 2010 Pearson Prentice Hall. All rights reserved 5 -2
Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty. © 2010 Pearson Prentice Hall. All rights reserved 5 -3
The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. © 2010 Pearson Prentice Hall. All rights reserved 5 -4
In probability, an experiment is any process that can be repeated in which the results are uncertain. The sample space, S, of a probability experiment is the collection of all possible outcomes. An event is any collection of outcomes from a probability experiment. An event may consist of one outcome or more than one outcome. In general, events are denoted using capital letters such as E. © 2010 Pearson Prentice Hall. All rights reserved 5 -5
EXAMPLE Identifying Events and the Sample Space of a Probability Experiment Consider the probability experiment of having two children. (a) Determine the sample space. (b) Define the event E = “have one boy”. (a) {(boy, boy), (boy, girl), (girl, boy), (girl, girl)} (b) {(boy, girl), (girl, boy)} © 2010 Pearson Prentice Hall. All rights reserved 5 -6
© 2010 Pearson Prentice Hall. All rights reserved 5 -7
© 2010 Pearson Prentice Hall. All rights reserved 5 -8
EXAMPLE A Probability Model In a bag of peanut M&M milk chocolate candies, the colors of the candies can be brown, yellow, red, blue, orange, or green. Suppose that a candy is randomly selected from a bag. The table shows each color and the probability of drawing that color. Verify this is a probability model. Color Probability Brown 0. 12 Yellow 0. 15 Red 0. 12 Blue 0. 23 Orange 0. 23 Green 0. 15 • All probabilities are between 0 and 1, inclusive. • Because 0. 12 + 0. 15 + 0. 12 + 0. 23 + 0. 15 = 1, rule 2 (the sum of all probabilities must equal 1) is satisfied. © 2010 Pearson Prentice Hall. All rights reserved 5 -9
If an event is impossible, the probability of the event is 0. If an event is a certainty, the probability of the event is 1. An unusual event is an event that has a low probability of occurring; that is, P(E)<0. 05 © 2010 Pearson Prentice Hall. All rights reserved 5 -10
© 2010 Pearson Prentice Hall. All rights reserved 5 -11
© 2010 Pearson Prentice Hall. All rights reserved 5 -12
EXAMPLE Building a Probability Model Theoretically, 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 1/36 =. 028 2/36 =. 056 3/36 =. 083 4/36 =. 111 5/36 =. 139 6/36 =. 167 5/36 =. 139 4/36 =. 111 3/36 =. 083 2/36 =. 056 1/36 =. 028 250 rolls of two six-sided dice. 2 6 (. 024) 3 10 (. 04) 4 25 (. 10) 5 35 (. 14) 6 35 (. 14) 7 39 (. 156) 8 38 (. 152) 9 34 (. 136) 10 10 (. 04) 11 10 (. 04) 12 8 (. 032) 5 -13
© 2010 Pearson Prentice Hall. All rights reserved 5 -14
The classical method of computing probabilities requires equally likely outcomes. An experiment is said to have equally likely outcomes when each simple event has the same probability of occurring. © 2010 Pearson Prentice Hall. All rights reserved 5 -15
EXAMPLE Computing Probabilities Using the Classical Method Suppose a “fun size” bag of M&Ms contains 9 brown candies, 6 yellow candies, 7 red candies, 4 orange candies, 2 blue candies, and 2 green candies. Suppose that a candy is randomly selected. (a) What is the probability that it is yellow? (b) What is the probability that it is blue? (c) Comment on the likelihood of the candy being yellow versus blue. (a) There a total of 9 + 6 + 7 + 4 + 2 = 30 candies, so N(S) = 30. (b) P(blue) = 2/30 = 0. 067. (c) Since P(yellow) = 6/30 and P(blue) = 2/30, selecting a yellow is three times as likely as selecting a blue. © 2010 Pearson Prentice Hall. All rights reserved 5 -16
© 2010 Pearson Prentice Hall. All rights reserved 5 -17
EXAMPLE Using Simulation Use the probability applet to simulate throwing a 6 sided die 100 times. Approximate the probability of rolling a 4. How does this compare to the classical probability? Repeat the exercise for 1000 throws of the die. © 2010 Pearson Prentice Hall. All rights reserved 5 -18
© 2010 Pearson Prentice Hall. All rights reserved 5 -19
The subjective probability of an outcome is a probability obtained on the basis of personal judgment. For example, an economist predicting there is a 20% chance of recession next year would be a subjective probability. © 2010 Pearson Prentice Hall. All rights reserved 5 -20
Section 5. 2 Probability Rules © 2010 Pearson Prentice Hall. All rights reserved 5 -21
© 2010 Pearson Prentice Hall. All rights reserved 5 -22
Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events. © 2010 Pearson Prentice Hall. All rights reserved 5 -23
We often draw pictures of events using Venn diagrams. These pictures represent events as circles enclosed in a rectangle. The rectangle represents the sample space, and each circle represents an event. For example, suppose we randomly select a chip from a bag where each chip in the bag is labeled 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Let E represent the event “choose a number less than or equal to 2, ” and let F represent the event “choose a number greater than or equal to 8. ” These events are disjoint as shown in the figure. © 2010 Pearson Prentice Hall. All rights reserved 5 -24
© 2010 Pearson Prentice Hall. All rights reserved 5 -25
EXAMPLE The Addition Rule for Disjoint Events The probability model to the right shows the distribution of the number of rooms in housing units in the United States. Number of Rooms in Housing Unit Probability One 0. 010 Two 0. 032 Three 0. 093 Four 0. 176 Five 0. 219 Six 0. 189 All probabilities are between 0 and 1, inclusive. Seven 0. 122 Eight 0. 079 0. 010 + 0. 032 + … + 0. 080 = 1 Nine or more 0. 080 (a) Verify that this is a probability model. Source: American Community Survey, U. S. Census Bureau © 2010 Pearson Prentice Hall. All rights reserved 5 -26
Number of Rooms in Housing Unit Probability (b) What is the probability a randomly selected housing unit has two or three rooms? One 0. 010 Two 0. 032 Three 0. 093 Four 0. 176 = P(two) + P(three) Five 0. 219 = 0. 032 + 0. 093 Six 0. 189 = 0. 125 Seven 0. 122 Eight 0. 079 Nine or more 0. 080 P(two or three) © 2010 Pearson Prentice Hall. All rights reserved 5 -27
Number of Rooms in Housing Unit Probability (c) What is the probability a randomly selected housing unit has one or two or three rooms? One 0. 010 Two 0. 032 Three 0. 093 Four 0. 176 P(one or two or three) Five 0. 219 = P(one) + P(two) + P(three) Six 0. 189 Seven 0. 122 = 0. 010 + 0. 032 + 0. 093 Eight 0. 079 Nine or more 0. 080 = 0. 135 © 2010 Pearson Prentice Hall. All rights reserved 5 -28
© 2010 Pearson Prentice Hall. All rights reserved 5 -29
© 2010 Pearson Prentice Hall. All rights reserved 5 -30
EXAMPLE Illustrating the General Addition Rule Suppose that a pair of dice are thrown. Let E = “the first die is a two” and let F = “the sum of the dice is less than or equal to 5”. Find P(E or F) using the General Addition Rule. © 2010 Pearson Prentice Hall. All rights reserved 5 -31
(2, 4)(2, 5), (2, 6) (2, 1) (2, 2) (2, 3) (1, 1), (1, 2) (1, 3), (1, 4) (3, 1), (3, 2) (4, 1) © 2010 Pearson Prentice Hall. All rights reserved 5 -32
© 2010 Pearson Prentice Hall. All rights reserved 5 -33
Complement of an Event Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted EC is all outcomes in the sample space S that are not outcomes in the event E. © 2010 Pearson Prentice Hall. All rights reserved 5 -34
Complement Rule If E represents any event and EC represents the complement of E, then P(EC) = 1 – P(E) © 2010 Pearson Prentice Hall. All rights reserved 5 -35
EXAMPLE Illustrating the Complement Rule According to the American Veterinary Medical Association, 31. 6% of American households own a dog. What is the probability that a randomly selected household does not own a dog? P(do not own a dog) = 1 – P(own a dog) = 1 – 0. 316 = 0. 684 © 2010 Pearson Prentice Hall. All rights reserved 5 -36
EXAMPLE Computing Probabilities Using Complements The data to the right represent the travel time to work for residents of Hartford County, CT. (a) What is the probability a randomly selected resident has a travel time of 90 or more minutes? There a total of 24, 358 + 39, 112 + … + 4, 895 = 393, 186 residents in Hartford County, CT. The probability a randomly selected resident will have a commute time of “ 90 or more minutes” is Source: United States Census Bureau © 2010 Pearson Prentice Hall. All rights reserved 5 -37
(b) Compute the probability that a randomly selected resident of Hartford County, CT will have a commute time less than 90 minutes. P(less than 90 minutes) = 1 – P(90 minutes or more) = 1 – 0. 012 = 0. 988 © 2010 Pearson Prentice Hall. All rights reserved 5 -38
Section 5. 3 Independence and Multiplication Rule © 2010 Pearson Prentice Hall. All rights reserved 5 -39
© 2010 Pearson Prentice Hall. All rights reserved 5 -40
Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F. © 2010 Pearson Prentice Hall. All rights reserved 5 -41
EXAMPLE Independent or Not? (a) Suppose you draw a card from a standard card deck and then draw another card without replacement. The events “draw an ace” and “draw another ace” are dependent because the result of the first draw impacted the result of the second draw. (b) Suppose you draw a card from a standard card deck and then draw another card with replacement. The events “draw an ace” and “draw another ace” are independent because the result of choosing a card did not impact the result of the second draw. (c) The events “it will rain tomorrow” and “I play tennis tomorrow” are dependent events. d) The events “the price of gasoline goes down” and “John passes Indrihovic’s Statistics Course” are independent events. © 2010 Pearson Prentice Hall. All rights reserved 5 -42
© 2010 Pearson Prentice Hall. All rights reserved 5 -43
© 2010 Pearson Prentice Hall. All rights reserved 5 -44
EXAMPLE Computing Probabilities of Independent Events The probability that a randomly selected female aged 60 years old will survive the year is 99. 186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that two randomly selected 60 year old females will survive the year? The survival of the first female is independent of the survival of the second female. We also have that P(survive) = 0. 99186. 5 -45
EXAMPLE Computing Probabilities of Independent Events A manufacturer of exercise equipment knows that 10% of their products are defective. They also know that only 30% of their customers will actually use the equipment in the first year after it is purchased. If there is a one-year warranty on the equipment, what proportion of the customers will actually make a valid warranty claim? We assume that the defectiveness of the equipment is independent of the use of the equipment. So, © 2010 Pearson Prentice Hall. All rights reserved 5 -46
© 2010 Pearson Prentice Hall. All rights reserved 5 -47
EXAMPLE Illustrating the Multiplication Principle for Independent Events The probability that a randomly selected female aged 60 years old will survive the year is 99. 186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that four randomly selected 60 year old females will survive the year? P(all four survive) = P (1 st survives and 2 nd survives and 3 rd survives and 4 th survives) = P(1 st survives). P(2 nd survives). P(3 rd survives). P(4 th survives) = (0. 99186) = 0. 9678 © 2010 Pearson Prentice Hall. All rights reserved 5 -48
© 2010 Pearson Prentice Hall. All rights reserved 5 -49
EXAMPLE Computing “at least” Probabilities The probability that a randomly selected female aged 60 years old will survive the year is 99. 186% according to the National Vital Statistics Report, Vol. 47, No. 28. What is the probability that at least one of 500 randomly selected 60 year old females will die during the course of the year? P(at least one dies) = 1 – P(none die) = 1 – P(all survive) = 1 – 0. 99186500 = 0. 9832 © 2010 Pearson Prentice Hall. All rights reserved 5 -50
Section 5. 4 Conditional Probability and the General Multiplication Rule © 2010 Pearson Prentice Hall. All rights reserved 5 -51
© 2010 Pearson Prentice Hall. All rights reserved 5 -52
Conditional Probability The notation P(F | E) is read “the probability of event F occurring on the condition that event E has already occurred. ” Essentially, this reduces the sample space to event E. © 2010 Pearson Prentice Hall. All rights reserved 5 -53
EXAMPLE An Introduction to Conditional Probability Suppose that a single six-sided die is rolled. What is the probability that the die is an odd number, P(O)? Now suppose that the die is rolled a second time, but we are told the outcome will be a prime number, Event P. What is the probability that the die is an odd number? - that is, find P(F |E). First roll: S = {1, 2, 3, 4, 5, 6} Second roll: S P = {2, 3, 5} O P 2 3, 5 4, 6 1 Note, event P is a subset of S. Once P occurs, the sample space is reduced to set P. © 2010 Pearson Prentice Hall. All rights reserved 5 -54
© 2010 Pearson Prentice Hall. All rights reserved 5 -55
EXAMPLE Conditional Probabilities on Belief about God and Region of the Country A survey was conducted by the Gallup Organization conducted May 8 – 11, 2008 in which 1, 017 adult Americans were asked, “Which of the following statements comes closest to your belief about God – you believe in the biblical God, you don’t believe in the Biblical God, but you do believe in a universal spirit or higher power, or you don’t believe in either? ” The results of the survey, by region of the country, are given in the table below. Believe in a Don’t believe Biblical God universal spirit in either East 204 36 15 Midwest 212 29 13 South 219 26 9 West 152 76 26 (a) What is the probability that a randomly selected adult American who lives in the East believes in a Biblical God? (b) What is the probability that a randomly selected adult American who believes in the Biblical God lives in the East? © 2010 Pearson Prentice Hall. All rights reserved 5 -56
Believe in God Believe in universal spirit Don’t believe in either East 204 36 15 Midwest 212 29 13 South 219 26 9 West 152 76 26 (a) What is the probability that a randomly selected adult American who lives in the East believes in biblical God? (b) What is the probability that a randomly selected adult American who believes in biblical God lives in the East? © 2010 Pearson Prentice Hall. All rights reserved 5 -57
EXAMPLE Murder Victims In 2005, 19. 1% of all murder victims were between the ages of 20 and 24 years old. Also in 2005, 16. 6% of all murder victims were 20 – 24 year old males. What is the probability that a randomly selected murder victim in 2005 was male given that the victim is 20 - 24 years old? = 0. 869 y 20 to 24. 025 Male. 166 x Note: x is the probability of being a male murder victim over 24 years old. y is the probability of being a female murder victim over 24 years old. 0. 025 is the probability of being a female murder victim between 20 and 24 years old. © 2010 Pearson Prentice Hall. All rights reserved 5 -58
Section 5. 5 Counting Techniques © 2010 Pearson Prentice Hall. All rights reserved 5 -59
© 2010 Pearson Prentice Hall. All rights reserved 5 -60
A combination is an arrangement, without regard to order, of n distinct objects without repetitions. The symbol n. Cr represents the number of combinations of n distinct objects taken r at a time, where r < n. © 2010 Pearson Prentice Hall. All rights reserved 5 -61
© 2010 Pearson Prentice Hall. All rights reserved 5 -62
Determine the value of 9 C 3. © 2010 Pearson Prentice Hall. All rights reserved 5 -63
The United States Senate consists of 100 members. In how many ways can 4 members be randomly selected to attend a luncheon at the White House? © 2010 Pearson Prentice Hall. All rights reserved 5 -64
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