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Chapter 6

Chapter 6. Probability. Chance experiment – any activity or situation in which there is uncertainty about which of two or more plausible outcomes will result. Suppose two six-sided die is rolled and they both land on sixes. Or a coin is flipped and it lands on heads.

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Chapter 6

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  1. Chapter 6 Probability

  2. Chance experiment– any activity or situation in which there is uncertainty about which of two or more plausible outcomes will result. Suppose two six-sided die is rolled and they both land on sixes. Or a coin is flipped and it lands on heads. Or record the color of the next 20 cars to pass an intersection. These would be examples of chance experiments.

  3. Sample space- the collection of all possible outcomes of a chance experiment Suppose a six-sided die is rolled. The possible outcomes are that the die could land with 1 dot up or 2, 3, 4, 5, or 6 dots up. S = {1, 2, 3, 4, 5, 6} This would be an example of a sample space. The sum of the probabilities of the outcomes in the sample space equals ONE. “S” stands for sample space. We use set notation to list the outcomes of the sample space.

  4. Suppose two coins are flipped. The sample space would be: S = {HH, HT, TH, TT} Where H = heads and T = tails We can also use a tree diagram to represent a sample space. H H We follow the branches out to show an outcome. HT T H T T

  5. Suppose a six-sided die is rolled. The outcome that the die would land on an even number would be E = {2, 4, 6} This would be an example of an event. Event-any collection of outcomes (subset) from the sample space of a chance experiment We typically use capital letters to denote an event.

  6. Complement - Consists of all outcomes that are not in the event E’ and E also denote the complement of E Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} What would the event be that is the die NOT landing on an even number? EC = {1, 3, 5} This is an example of complementary events. The sum of the probabilities of complementary events equals ONE. The superscript “C” stands for complement

  7. These complementary events can be shown on aVenn Diagram. E = {2, 4, 6} and EC = {1, 3, 5} Let the circle represent event E. Let the rectangle represent the sample space. Let the shaded area represent event not E.

  8. Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} The event that the die would land on a prime number would be P = {2, 3, 5} What would be the event E or P happening? E or P = {2, 3, 4, 5, 6} This is an example of the union of two events.

  9. The union of A or B - consists of all outcomes that are in at least one of the two events, that is, in A or in B or in both. Consider a marriage or union of two people – when two people marry, what do they do with their possessions ? This symbol means “union” The bride takes all her stuff & the groom takes all his stuff & they put it together! And live happily ever after! This is similar to the union of A and B. All of A and all of B are put together!

  10. Let’s revisit rolling a die and getting an even or a prime number . . . E or P = {2, 3, 4, 5, 6} Another way to represent this is with a Venn Diagram. E or P would be any number in either circle. Even number Prime number Why is the number 1 outside the circles? 3 4 2 6 5 1

  11. Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} The event that the die would land on a prime number would be P = {2, 3, 5} What would be the event E andP happening? E and P = {2} This is an example of the intersection of two events.

  12. The intersection of A and B - consists of all outcomes that are in both of the events This symbol means “intersection”

  13. Let’s revisit rolling a die and getting an even or a prime number . . . E and P = {2} To represent this with a Venn Diagram: E and P would be ONLY the middle part that the circles have in common 3 4 2 6 5 1

  14. Mutually exclusive(or disjoint) events -two events have no outcomes in common; two events that NEVER happen simultaneously Suppose a six-sided die is rolled. Consider the following 2 events: A = {2} B = {6} On a single die roll, is it possible for A and B to happen at the same time? These events are mutually exclusive.

  15. A Venn Diagram for the roll of a six-sided die and the following two events: A = {2} B = {6} A and B are mutually exclusive (disjoint) since they have no outcomes in common The intersection of A and B is empty! 3 4 6 2 1 5

  16. Practice with Venn Diagrams On the following four slides you will find Venn Diagrams representing the students at your school. Some students are enrolled inStatistics,some inCalculus,and some inComputer Science. For the next four slides, indicate what relationships the shaded regions represent. (use complement, intersection, and union)

  17. Statistics Calculus Computer Science Calculus or Computer Science

  18. Statistics Calculus Computer Science (Statistics or Computer Science) and not Calculus

  19. Statistics Calculus Computer Science Com Sci Statistics and Computer Science and not Calculus

  20. Statistics Calculus Computer Science Statistics and not (Computer Science or Calculus)

  21. What is Probability? Three different approaches to probability

  22. The Classical Approach When the outcomes in a sample space are equallylikely, the probability of an event E, denoted by P(E), is the ratio of the number of outcomes favorable to E to the total number of outcomes in the sample space. Examples: flipping a coin, rolling a die, etc.

  23. On some football teams, the honor of calling the toss at the beginning of the football game is determined by random selection. Suppose this week a member of the 11-player offensive team will be selected to call the toss. There are five interior linemen on the offensive team. If event L is defined as the event that an interior linemen is selected to call the toss, what is probability of L? P(L) = 5/11

  24. Consider an archer shooting arrows at a target. The probability of getting a bulls’ eye should be the ratio of the area of the inner circle to the area of the entire target. What if a very experienced archer were shooting the arrows? Would the probability of a bull’s eye still be the same? The classical approach doesn’t work for every situation.

  25. The Relative Frequency Approach The probability of event E, denoted by P(E), is defined to be the value approached by the relative frequency of occurrence of E in a very long series of trials of a chance experiment. Thus, if the number of trials is quite large,

  26. Consider flipping a coin and recording the relative frequency of heads. When the number of coin flips is small, there is a lot of variability in the relative frequency of “heads” (as shown in this graph). What do you notice in the graph at the right?

  27. Consider flipping a coin and recording the relative frequency of heads. The graph at the right shows the relative frequency when the coin is flipped a large number of times. What do you notice in this graph at the right?

  28. Law of Large Numbers Notice how the relative frequency of heads approaches ½ the larger the number of trials! As the number of repetitions of a chance experiment increase, the chance thatthe relative frequency of occurrence for an event will differ from the true probability by more than any small number approaches 0. OR in other words, after a large number of trials, the relative frequency approaches the true probability.

  29. The Subjective Approach Probability can be interpreted as a personal measure of the strength of belief that a particular outcome will occur. Theproblem with a subjective approach is that different people could assigndifferentprobabilitiesto thesame outcomebased on their subjective viewpoints. Example: An airline passenger may report that her probability of being placed on standby (denied a seat) due to overbooking is 0.1. She arrived at this through personal experience and observation of events.

  30. Probability Rules!

  31. Property 1.Legitimate Values For any event E, 0 <P(E) < 1 Property 2.Sample space If S is the sample space, P(S) = 1 Fundamental Properties of Probability

  32. Properties Continued . . . Property 3. Addition If two events E and F are disjoint,P(E or F) = P(E) + P(F) Property 4.Complement For any event E, P(E) + P(not E) = 1

  33. Probabilities of Equally Likely Outcomes Consider an experiment that can result in any one of N possible outcomes. Denote the simple events by O1, O2, …, ON. If these simple events are equally likely to occur, then 1. 2. For any event E,

  34. Suppose you roll a six-sided die once. Let E be the event that you roll an even number. P(E) = P(2 or 4 or 6) = 3/6 Number of outcomes in E Over N

  35. Addition Rule for Disjoint Events If events E1, E2, . . ., Ek are disjoint (mutually exclusive) events, then P(E1 or E2 or . . . or Ek) = P(E1) + P(E2) + . . . + P(Ek) In words, the probability that any of these k disjoint events occurs is the sum of the probabilities of the individual events.

  36. A large auto center sells cars made by many different manufacturers. Three of these are Honda, Nissan, and Toyota. Consider a chance experiment that consist of observing the make of the next car sold. Suppose that P(H) = 0.25, P(N) = 0.18, P(T) = 0.14. Are these disjoint events? yes P(H or N or T) = .25 + .18+ .14 = .57 P(not (H or N or T)) = 1 - .57 = .43

  37. Sometimes the knowledge that one event has occurred changes our assessment of the likelihood that another event occurs. Consider the following example: Suppose that 0.1% of all the individuals in a population have a certain disease. The presence of the disease is not discernable from appearances, but there is a screening test for the disease. Let D = the event that a person has the disease P(D) = .001

  38. Disease example continued . . . Suppose that 0.1% of all the individuals in a population have a certain disease. 80% of those with positive test results actually have the disease. 20% of those with positive test results actually do NOThave the disease (false positive) Let P = the event that a person tests positive for the disease P(D|P) = 0.80 This is an example of conditional probability. Knowing that event P, the person tested positive, has occurred, changes the probability of event D, the person having the disease, from 0.001 to 0.80. Read: Probability that a person has the disease “GIVEN” the person tests positive

  39. A probability that takes into account a given condition has occurred Conditional Probability

  40. The article “Chances Are You Know Someone with a Tattoo, and He’s Not a Sailor”(Associated Press, June 11, 2006) included results from a survey of adults aged 18 to 50. The accompanying data are consistent with summary values given in the article. Assuming these data are representative of adult Americans and that an adult is selected at random, use the given information to estimate the following probabilities.

  41. Tattoo Example Continued . . . What is the probability that a randomly selected adult has a tattoo? P(tattoo) = 24/100 = 0.24

  42. Tattoo Example Continued . . . This is a condition! How many adults in the sample are ages 18-29? How many adults in the sample are ages 18-29 AND have a tattoo? What is the probability that a randomly selected adult has a tattoo if they are between 18 and 29 years old? P(tattoo|age 18-29) = 18/50 = 0.36

  43. Tattoo Example Continued . . . How many adults in the sample have a tattoo? How many adults in the sample are ages 18-29 AND have a tattoo? What is the probability that a randomly selected adult is between 18 and 29 years old if they have a tattoo? This is a condition! P(age 18-29|tattoo) = 18/24 = 0.75

  44. Sometimes the knowledge that one event has occurred does NOT change our assessment of the likelihood that another event occurs. Consider the genetic trait, hitch hiker’s thumb, which is the ability to bend the last joint of the thumb back at an angle of 60° or more. Whether or not an offspring has hitch hiker’s thumb is determined by two random events: which gene is contributed by the father and which gene is contributed by the mother. Which gene is contributed by the father does NOT affect which gene is contributed by the mother These are independent events.

  45. Let’s consider a bank that offers different types of loans: The bank offers both adjustable-rate and fixed-rate loans on single-family dwellings, condominiums and multifamily dwellings. The following table, called a joint-probability table, displays probabilities based upon the bank’s long-run loaning practices. P(Adjustable loan) = .70

  46. Bank Loan’s Continued . . . P(Adjustable loan) = .70 P(Adjustable loan|Condo) = .21/.30 = .70 Knowing that the loan is for a condominium does not change the probability that it is an adjustable-rate loan. Therefore, the event that a randomly selected loan is adjustable and the event that a randomly selected loan is for a condo are independent.

  47. Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs. Two events, E and F, are said to be independent if P(E|F) = P(E). If P(E|F) = P(E), it is also true that P(F|E) = P(F). Independent Events If two events are not independent, they are said to be dependent events.

  48. Multiplication Rule for Two Independent Events Two events E and F are independent, if and only if,

  49. Hitch Hiker’s Thumb Revisited Suppose that there is a 0.10 probability that a parent will pass along the hitch hiker’s thumb gene to their offspring. What is the probability that a child will have a hitch hiker’s thumb? Since these are independent events, we just multiply the probabilities together. This would happen if the mother contributes a hitch hiker’s gene (H+) AND if the father contributes a hitch hiker’s gene (H+). P(H+ from mom AND H+ from dad) = 0.1 × 0.1 = 0.01

  50. Multiplication Rule for k Independent Events Events E1, E2, . . ., Ek are independent if knowledge that some number of the events have occurred does not change the probabilities that any particular one or more of the other events occurred. This relationship remains valid if one or more of the events are replaced by their complement (not E).

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