Chapter 4 Probability

Chapter 4Probability 4-1 Review and Preview 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Counting

Section 4-1 Review and Preview

Review Necessity of sound sampling methods. Common measures of characteristics of data Mean Standard deviation

Preview 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. Statisticians use the rare event rule for inferential statistics.

Section 4-2 Basic Concepts of Probability

Part 1 Basics of Probability

Events and Sample Space • Event any collection of results or outcomes of a procedure • Simple Event an outcome or an event that cannot be further broken down into simpler components • Sample Space for a procedure consists of all possible simple events; that is, the sample space consists of all outcomes that cannot be broken down any further

Example • A pair of dice are rolled. The sample space has 36 simple events: • 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. • Which elements of the sample space correspond to the event that the sum of each dice is 4?

Example • Which elements of the sample space correspond to the event that the sum of each dice is 4? • ANSWER: • 3,1 2,2 1,3

P- denotes a probability. A, B, andC- denote specific events. P(A)- denotes the probability of event A occurring. Notation for Probabilities

Rule 1: Relative Frequency Approximation of Probability Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based on these actual results, P(A) is approximated as follows: P(A)= # of times A occurred # of times procedure was repeated Basic Rules for Computing Probability

Problem 20 on page 149 F = event of a false negative on polygraph test Thus this is not considered unusual since it is more than 0.001 (see page 146). The test is not highly accurate. Example

Rounding Off Probabilities When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. All digits are significant except for the zeros that are included for proper placement of the decimal point. Example: 0.1254 has four significant digits 0.0013 has two significant digits

Problem 21 on page 149 F = event of a selecting a female senator NOTE: total number of senators=100 Thus this does not agree with the claim that men and women have an equal (50%) chance of being selected as a senator. Example

Problem 28 on page 150 A = event that Delta airlines passenger is involuntarily bumped from a flight Thus this is considered unusual since it is less than 0.05 (see directions on page 149). Since probability is very low, getting bumped from a flight on Delta is not a serious problem. Example

Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes) Assume that for a given procedure each simple event has an equal chance of occurring. Basic Rules for Computing Probability P(A) = number of ways A can occur number of different simple events in the sample space

What is the probability of rolling two die and getting a sum of 4? A = event that sum of the dice is 4 Assume each number is equallylikely to be rolled on the die. Rolling a sum of 4 can happen in one of three ways (see previous slide) with 36 simple events so: Example

What is the probability of getting no heads when a “fair” coin is tossed three times? (A fair coin has an equal probability of showing heads or tails when tossed.) A = event that no heads occurs in three tosses Sample Space (in order of toss): Example

Sample space has 8 simple events. Event A corresponds to TTT only so that: Example

Problem 36 on page 151 Let: S = event that son inherits disease (xY or Yx) D = event that daughter inherits disease (xx) Example

Problem 36 on page 151 (a) Father: xY Mother: XX Sample space for a son: YX YX Sample space has no simple events that represent a son that has the disease so: Example

Problem 36 on page 151 (b) Father: xY Mother: XX Sample space for a daughter: xX xX Sample space has no simple events that represent a daughter that has the disease so: Example

Problem 36 on page 151 (c) Father: XY Mother: xX Sample space for a son: Yx YX Sample space has one simple event that represents a son that has the disease so: Example

Problem 36 on page 151 (d) Father: XY Mother: xX Sample space for a daughter: Xx XX Sample space has no simple event that represents a daughter that has the disease so: Example

Problem 18 on page 149 Table 4-1 on page 137 (polygraph data) Example

It is helpful to first total the data in the table: (a) How many responses were lies: ANSWER: 51 Example

If one response is randomly selected, what is the probability it is a lie? L = event of selecting one of the lie responses (c) Example

Rule 3: Subjective Probabilities Basic Rules for Computing Probability - continued P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.

Problem 4 on page 148 Probability should be high based on experience (it is rare to be delayed because of an accident). Guess: P=99/100 (99 out of 100 times you will not be delayed because of an accident) ANSWERS WILL VARY Example

Example: Classical probability predicts the probability of flipping a (non-biased) coin and it coming up heads is ½=0.5 Ten coin flips will sometimes result in exactly 5 heads and a frequency probability of heads 5/10=0.5; but often you will not get exactly 5 heads in ten flips.

As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability. Example: If we flip a coin 1 million times the frequency probability should be approximately 0.5 Law of Large Numbers

Probability Limits The probability of an event that is certain to occur is 1. Always express a probability as a fraction or decimal number between 0 and 1. • The probability of an impossible event is 0. • For any event A, the probability of A is between 0 and 1 inclusive. That is: 0 P(A)  1

Possible Values for Probabilities

The complement of event A, denoted by A, consists of all outcomes in which the event A does not occur. Complementary Events

If a fair coin is tossed three times and A = event that exactly one heads occurs Find the complement of A. Example

Sample space: Event A corresponds to HTT, THT, TTH Therefore, the complement of A are the simple events: HHH, HHT, HTH, THH, TTT Example

Part 2 Beyond theBasics of Probability: Odds

Odds The actual odds in favor of event A occurring are the ratio P(A)/ P(A), usually expressed in the form of a:b (or “a to b”), where a and b are integers having no common factors. The actual odds against event A occurring are the ratio P(A)/P(A), which is the reciprocal of the actual odds in favor of the event. If the odds in favor of A are a:b, then the odds against A are b:a. The payoff odds against event A occurring are the ratio of the net profit (if you win) to the amount bet. payoff odds against event A = (net profit) : (amount bet)

Example Problem 38, page 149 W = simple event that you win due to an odd number Sample Space 00, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 (a) There are 18 odd numbers so that P(W) = 18/38

Example Problem 38, page 149 • There are 20 events that correspond to winning from a number that is not odd (i.e. you do not win due to an odd number) so: • (b) Odds against winning are

Example Problem 38, page 149 • (c) Payoff odds against winning are 1:1 • That is, $1 net profit for every $1 bet • Thus, if you bet $18 and win, your net profit is $18 which can be found by solving the proportion: • The casino returns $18+$18=$36 to you.

Example Problem 38, page 149 • (d) Actual odds against winning are 10:9 • That is, $10 net profit for every $9 bet • Thus, if you bet $18 and win, your net profit is $20 which can be found by solving the proportion: • The casino returns $18+$20=$38 to you.

Recap In this section we have discussed: • Rare event rule for inferential statistics. • Probability rules. • Law of large numbers. • Complementary events. • Rounding off probabilities. • Odds.

Section 4-3 Addition Rule

Key Concept This section presents the addition rule as a device for finding probabilities that can be expressed as P(A or B), the probability that either event A occurs or event B occurs (or they both occur) as the single outcome of the procedure. The key word in this section is “or.” It is the inclusive or, which means either one or the other or both.

Compound Event any event combining 2 or more simple events Compound Event Notation P(A or B) = P (in a single trial, event A occurs or event B occurs or they both occur)

When finding the probability that event A occurs or event B occurs, find the total number of ways A can occur and the number of ways B can occur, but find that total in such a way that no outcome is counted more than once. General Rule for a Compound Event

A random survey of members of the class of 2005 finds the following: What is the probability the student did not graduate or was a man? Example

Method 1: directly add up those who did not graduate and those who are men (without counting men twice): 22 + 19 + 582 = 623 Method 2: add up the total number who did not graduate and the total number of men, then subtract the double count of men: 41 + 601 - 19 = 623 Example

The probability the student did not graduate or was a man : 623/1295 = 0.481 Example

Chapter 4 Probability