ELEMENTARY STATISTICS Chapter 3 Probability MARIO F TRIOLA

Chapter 3 Probability 3 -1 Overview 3 -2 Fundamentals 3 -7 Counting 3 -3 Addition Rule 3 -4 Multiplication Rule: Basics 3 -5 Multiplication Rule: Complements and Conditional Probability 3 -6 Probabilities Through Simulations Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 2

3 -1 Overview Objectives v develop sound understanding of probability values used in subsequent chapters v develop basic skills necessary to solve simple probability problems Rare Event Rule for Inferential Statistics: If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 3

3 -2 Probability Fundamentals Definitions v. Event - any collection of results or outcomes from some procedure. v Simple event - any outcome or event that cannot be broken down into simpler components. v Sample space - all possible simple events. Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4

Notation P - denotes a probability A, B, . . . - denote specific events P (A) - denotes the probability of event A occurring Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5

Basic Rules for Computing Probability Rule 1: Relative Frequency Approximation Conduct (or observe) an experiment a large number of times, and count the number of times event A actually occurs, then an estimate of P(A) is P(A) = number of times A occurred number of times trial was repeated Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6

Basic Rules for Computing Probability Rule 2: Classical approach (requires equally likely outcomes) If a procedure has n different simple events, each with an equal chance of occurring, and s is the number of ways event A can occur, then s P(A) = number of ways A can occur number of different simple events Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7

Basic Rules for Computing Probability Rule 3: Subjective Probabilities P(A), the probability of A, is found by simply guessing or estimating its value based on knowledge of the relevant circumstances. Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 8

Rule 1 The relative frequency approach is an approximation. Rule 2 The classical approach is the actual probability. Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9

Probability Limits v The probability of an impossible event is 0. v The probability of an event that is certain to occur is 1. 0 P(A) 1 Impossible to occur Certain to occur Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10

Possible Values for Probabilities 1 Certain Likely 0. 5 50 -50 Chance Unlikely Figure 3 -3 0 Impossible Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11

Example: Find the probability that a randomly selected person will be struck by lightning this year. The sample space consists of two simple events: either the person is struck by lightning or he/she is not. Because these simple events are not equally likely, we can use the relative frequency approximation (Rule 1) or subjectively estimate the probability (Rule 3). Using Rule 1, we can research past events to determine that in a recent year 377 people were struck by lightning in the US, which has a population of about 308, 400, 000. Therefore, P(struck by lightning in a year) 377 / 274, 037, 295 1/727, 000 Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 12

Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright

Example: On an ACT or SAT test, a typical multiplechoice question has 5 possible answers. If you make a random guess on one such question, what is the probability that your response is wrong? There are 5 possible outcomes or answers, and there are 4 ways to answer incorrectly. Random guessing implies that the outcomes in the sample space are equally likely, so we apply the classical approach (Rule 2) to get: P(wrong answer) = 4 / 5 = 0. 8 Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 14

Example: At Radnor High School, the use of cell phones in class is not allowed without teacher approval. However some students choose to ignore this rule. Based on your experience what is the probability that at least one person would break this rule in a randomly selected class? This event is not equally likely and there was no observations done. We are going to apply the subjective probability approach (Rule 3) to get: P(Breaking the rule) =. 90 Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 15

Law of Large Numbers As a procedure is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability. Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 16

Illustration of Law of Large Numbers Figure 3 -2 Chapter 3. Section 3 -1

Complementary Events The complement of event A, denoted by A, consists of all outcomes in which event A does not occur. P(A) (read “not A”) Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 18

Example: Testing Corvettes The General Motors Corporation wants to conduct a test of a new model of Corvette. A pool of 50 drivers has been recruited, 20 or whom are men. When the first person is selected from this pool, what is the probability of not getting a male driver? Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 19

Example: Testing Corvettes The General Motors Corporation wants to conduct a test of a new model of Corvette. A pool of 50 drivers has been recruited, 20 or whom are men. When the first person is selected from this pool, what is the probability of not getting a male driver? Because 20 of the 50 subjects are men, it follows that 30 of the 50 subjects are women so, P(not selecting a man) = P(woman) = 30 = 0. 6 50 Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 20

$Rounding Off Probabilities vgive the exact fraction or decimal or vround off the final$

Rounding Off Probabilities vgive the exact fraction or decimal or vround off the final result to three significant digits Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 21

Odds vactual odds against event A occurring are the ratio P(A), usually expressed in the form of a: b (or ‘a to b’), where a and b are integers with no common factors vactual odds in favor of event A are the reciprocal of the odds against that event, b: a (or ‘b to a’) Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 22

Odds v The payoff odds against event A represent the ratio of net profit (if you win) to the amount of the bet. Payoff odds against event A = (net profit): (amount bet) Chapter 3. Section 3 -1 and 3 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 23