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Chapter. 5. Probability. Section. 5.1. Probability Rules. Probability is a measure of the likelihood of an event occurring. Probability describes the long-term ratio with which a certain outcome will occur. ( Ex: ).

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  1. Chapter 5 Probability

  2. Chap 2

  3. Section 5.1 Probability Rules

  4. Probability is a measure of the likelihood of an event occurring. Probability describes the long-term ratio with which a certain outcome will occur. (Ex: ) Suppose you flipped a coin (H/T) 100 times. What proportion (ratio)of Heads would you expect to see? (in other words, make a prediction of the future)

  5. Probability deals with experiments (rolling a die or picking a card) that yield random results or outcomes. It reveals long-term predictability. If I rolled a die 1000 times, how many 5’s would I expect to see? since the Prob of getting a “5” is 1/6 = 0.167, if I conducted 1000 trials, I would expect to see a “5” approx 167 times.

  6. In probability, an experiment is any process/trial that can be repeated (picking a card) in which the outcome is uncertain. The sample space, S, of a probability experiment is the collection (set) of all possible outcomes. An eventis any collection of outcomes from a probability experiment and may consist of one, or more than one, outcome. Events are normally denoted using capital letters such as E. Events with only one possible outcome are called simple events,ei.

  7. EXAMPLE Identifying Events and the Sample Space of a Probability Experiment • Consider the probability experiment of having two children. • Identify all possible outcomes (Sample Space)of the probability experiment. • Define the event E = “have one boy”. • {(boy, boy), (boy, girl), (girl, boy), (girl, girl)} • E: (boy, girl), (girl, boy)}

  8. Rules of Probabilities (Very Important) 1. The probability of any event “E” occurring is written as P(E) where 0 ≤ P(E) ≤ 1. 2. The sum of the probabilities of all outcomes must equal 1. if S = {e1, e2, …, en}, thenP(e1) + P(e2) + … + P(en) = 1

  9. A probability modellists all possible outcomes of an experiment and each outcome’s probability of occurrence. It is also called a “Probability Distribution”, similar to a Frequency Distribution.

  10. EXAMPLE A Probability Distribution / Model In a bag of M&M candies, the colors can be brown, yellow, red, blue, orange, or green. Suppose that a candy is randomly drawn from the bag. This table shows each color and the probability of drawing that color. • Note that: • Each probability is between 0 and 1. • The sum of all probabilities equals 1

  11. If an event is impossible, then P(E) = 0 If an event is a certain, then P(E) = 1 If an event is unlikely to occur, then P(E) = 0.250 If an event is likely, then P(E) = 0.750

  12. “Empirical” Probability The probability of event E occurring is number of times E is actually observed, divided by the number of repetitions (trials) of the experiment. P(E)

  13. P(cruise) = Empirical Probability Empirical (or statistical) probability is based on observations The empirical frequency of event E is the relative frequency of event E. A travel agent observes that, of every 50 reservations she makes, 12 will be for a cruise. What is the probability that the next reservation she makes will be for a cruise?

  14. Empirical Probability… Suppose that one day I observed 852 cars pulling up to the Taco Bell order window. And of that number, I noted that 435 of those cars were white. Since 435/852 = 0.511, I would conclude that the probability is 0.511 that the next car to pull up to the window will be white. Or, I would predict that 511 of the next 1000 cars will be white. Chap 2

  15. 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.

  16. “Classical” Probability” If an experiment has “n” equally likely outcomes, and if the number of ways that an event “E” can occur is “m”, then:

  17. Classical Probability if S is the sample space of experiment “E”, where N(E) is the number of favorable outcomes in E, and N(S) is the total number of possible outcomes in the sample space.

  18. P(A) = “Probability of Event A.” Classical Probability Classical (or theoretical) probability is used when each outcome in a sample space is equally likely to occur. Can be calculated without need for actual observation. A die is rolled. Find the probability of Event A: rolling a 5. There are 6 possible outcomes but only one successful outcome in Event A and that is {5}

  19. Classical Probability… Suppose there are 52 playing cards in a deck, and 4 of those cards are “Aces”. Then, the Prob of my picking a card and having it be an Ace is: # of favorable outcomes / # of total outcomes Chap 2

  20. EXAMPLE Computing Probabilities Using the Classical Method Suppose a bag of M&M candies contains 9 brown, 6 yellow, 7 red, 4 orange, 2 blue, and 2 green candies. Now, suppose that a candy is randomly selected from this bag. (a) What is the probability that it is yellow? (b) What is the probability that it is blue? (c) What is the likelihood of the candy being yellow versus blue?

  21. EXAMPLE Using Simulation A “die” is a cube with “1 thru 6” numbers on each face. Use the probability applet in the TI-84 to simulate rolling a die 100 times and approximate the probability of throwing a “4”. Repeat the exercise for 1000 rolls of the die. Does this change the results you got with 100 trials? How does this compare to the classical probability of throwing a 4?

  22. The Law of Large Numbers As the number of trials of a probability experiment increases, the “empirical” proportion with which a certain outcome is actually observed approaches the theoretical probability of the outcome.

  23. Law of Large Numbers As an experiment is repeated a large number of times, the empirical probability of an event approaches the theoretical probability of the event.

  24. The “Subjective” or Intuitive probability of an outcome is a probability obtained purely on the basis of personal judgment. For example, an economist predicting there is a 20% chance of recession in the U.S. [ P(R) = 0.200]) next year is a subjective probability, also known as a “guess” or WAG.

  25. Section 5.2 The Addition Rule and Complements

  26. Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusiveevents. Ex: I pick one student from the room. The outcome of this event is either male or female, but if it is one, it cannot be the other.

  27. Disjoint Events If I select one student from the group, that student can be both a female and an adult (>21), but that student cannot be both a male and a female. So, E=female and E=adult are non-disjoint events, but E= female and E=male are disjoint events. Chap 2

  28. A and B A B A B Mutually Exclusive Events Two events, A and B, are mutually exclusive if they cannot occur at the same time. A and B are mutually exclusive. A and B are not mutually exclusive.

  29. A B 2 9 J 10 3 A 7 J J K 4 5 J 8 6 Q Mutually Exclusive Events Decide if the two events are mutually exclusive: Event A: Select a Jack from a deck of cards. Event B: Select a Heart from a deck of cards. Because the card can be a Jack and a Heart at the same time, the events are not mutually exclusive.

  30. Diagrams of events often use Venn diagrams. Events are circles enclosed in a rectangle which represents the sample space “S”. Suppose we randomly select a chip from a bag where each chip is numbered: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Now, let “E” represent event “choose a number less than or equal to 2,” and let “F” be “choose a number greater than or equal to 8.” These events are mutually exclusive (disjoint) as shown:

  31. Addition Rule for Disjoint Events If E and F are disjoint (mutually exclusive) events, then:

  32. The Addition Rule for Disjoint Events can be extended to more than two disjoint events. In general, if E, F, G, . . . each have no outcomes in common (disjoint), then:

  33. EXAMPLE The Addition Rule for Disjoint Events This distribution shows the probability of having “X” rooms in housing units in the United States. • Verify that this is a probability distribution model. • Each And ΣP = 1

  34. EXAMPLE The Addition Rule for Disjoint Events (b) What is the probability of randomly selected a housing unit that has two or three rooms? P(Two or Three) = P(Two) + P(Three) = 0.032 + 0.093 = 0.125

  35. EXAMPLE The Addition Rule for Disjoint Events (c) What is the probability a randomly selected housing unit has one or two or three rooms? P(one or two or three) = P(one) + P(two) + P(three) = 0.010 + 0.032 + 0.093 = 0.135

  36. General Addition Rule (works for all events, disjoint or not) For any two events E and F:

  37. The Addition Rule The probability that event A or B will occur is given by P (A or B) = P (A) + P (B) – P (A and B ) If events A and B are mutually exclusive, then the rule can be simplified to P (A or B) = P (A) + P (B). You roll a die. Find the probability that you roll either a number less than 3, or a 4. ( events are mutually exclusive ) P (roll a number less than 3 or roll a 4) = P (number is less than 3) + P (4)

  38. Addition Rule(Disjoint = Mutually Exclusive)

  39. EXAMPLE The General Addition Rule Suppose that a pair of dice are thrown. Let E = “the first die is a two” let F = “the sum of the two dice is less than or equal to 5”. Find P(E or F)

  40. = 0.361 Note: by subtracting the “3/36”, we have avoided counting these three outcomes (pairs of dice) twice.

  41. 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 S that are not outcomes of the event E.

  42. Complement Rule E represents any event EC represents the complement of E P(EC) = 1 – P(E)

  43. EXAMPLE Illustrating the Complement Rule According to the American Veterinary Assoc., 31.6% of American households own a dog. What is the probability that a randomly selected household does not own a dog? P(not dog) = 0.684

  44. 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 that a randomly selected resident has a travel time of 90 or more minutes? Source: United States Census Bureau

  45. EXAMPLE Computing Probabilities Using Complements There are: 24,358 + 39,112 + … + 4,895= 393,186 residents in Hartford County The probability that a randomly selected resident will have a commute time of 90 or more minutes is: Source: United States Census Bureau

  46. (b) If the Prob that a randomly selected resident of Hartford County has a commute time ≥ 90 min is 0.012, then what is the probability that a resident will have a commute time less than 90 minutes? P(<90 min) = 1 – 0.012 = 0.988

  47. Chap 2

  48. Section 5.3 Independent Events and the Multiplication Rule

  49. Multiplication Rule The probability that two events, A and B will occur in sequence is P (A and B) = P (A) · P (B | A). If event A and B are independent, then the rule can be simplified to P (A and B) = P (A) · P (B). Two cards are selected, without replacement, from a deck. Find the probability of selecting a diamond, and then selecting a spade. Because the card is not replaced, the events are dependent. P (diamondand spade) = P (diamond) · P (spade|diamond).

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