The Practice of Statistics Third Edition

Transcription

1 The Practice of Statistics Third Edition Chapter 6: Probability and Simulation: The Study of Randomness Copyright 2008 by W. H. Freeman & Company

2 Probability Rules True probability can only be found by many trials We need to describe probability mathematically. There are some facts that must be true for any assignment of probability. Probability is the long run proportion of repetitions on which an event occurs

3 In General Any probability is a number between zero and one. Zero means it never occurs, One means it always occurs The sum of the probabilities of all possible outcomes must equal one. Some outcome must occur on every trial

4 In General Continued If two events have no outcome in common, the probability that one or the other occurs is the sum of the individual events. The probability that an event does not occur is 1 minus the probability that the event does occur.

5 So, in summary:

6 Set Notation A U B A union B. Means A or B. A B A intersection B. Means A and B. Ø Empty Event (No outcomes) A B = Ø. This means A intersect B is empty. A and B are disjoint. It can be helpful to draw a picture. These are called Venn Diagrams.

7 Event A and Event B are disjoint because they do not overlap. They have no outcomes in common. Their union consists of the empty set.

8 This is a Venn Diagram of Event A and the complement A c A U A c is S A A c is the Empty Set (Ø).

9 Distance Learning Age Group 18 to to to & over Probability Think of our rules: Each probability is between 0 and 1 Probabilities total 1 What is the probability that the student we draw is not in the traditional age range (18 23) for an undergraduate? P(not 18 to 23) = 1 P(18 to 23 years) = =.43 Events 30 to 39 years and 40 & over are disjoint because no student can be in each group. P(30 or over) = P(30 to 39) + P(40 & over) = =.26

10 Playing Dice There are 36 possible outcomes when throwing two dice. That means each outcome has probability 1/36. What is the probability of rolling a 5? So 1/36 + 1/36 + 1/36 + 1/36 = 4/36 = 1/9 =.111 What is the probability of rolling a seven? It is the most common outcome of throwing two dice. 6/36 = 1/6 =.167

11 The two previous examples show one way to assign probabilities to events: Assign a probability to every individual event and add the probabilities to find the probability of any event. The probabilities have to sum to 1.

12 Benford s Law Faked numbers in tax returns, payment records, invoices and other business documents often display patterns that are not present in legitimate records. The first digit of numbers in legitimate records follow a pattern called Benford s Law First Digit Prob Let s go over the probabilities carefully.

13 Examples Event A = {First digit is 1} Event B = {First digit is 6 or greater} P(A) = P(1) =.301 P(B) = P(6) + P(7) + P(8) +P(9) = =.222 So P(A U B) = P(A) + P(B) = =.523 Remember the addition rule only applies to disjoint events!!!!!!

14 What Does This Mean Usually applies to the first digits of the sizes of similar quantities, such as invoices. Auditors can detect fraud by comparing the probabilities with the first digit in records such as invoices paid by businesses.

15 Equally Likely Outcomes You would think that the first digits would all be equally likely to show (1/9) If that was the case P(B) = P(6) + P(7) + P(8) +P(9) = 4/9 =.444 But actually P(B) =.222 So a crook who fakes random data has too many first digits 6 or greater and too few 1s and 2s.

16 There is a pretty simple rule for equally likely outcomes: Most random phenomena do not have equally likely outcomes, so this rule is not really that important.

17 Assignment Exercise 6.44 Read Pages Watch: