The Practice of Statistics Daniel S. Yates The Practice of Statistics Third Edition Chapter 6: Probability and Simulation: The Study of Randomness 6.2 Probability Models Copyright © 2008 by W. H. Freeman & Company

Essential Questions What is meant by a random phenomenon? How do you define probability in term of relative frequency? What is a sample space? How do you determine the number of outcomes in a sample space? What are the five rules for assignment of probability? What is meant by {A U B} and {A ∩ B}? How do you compute the probability of an event given the probabilities of outcomes that make up the event? How do you compute the probability of an event of equally likely outcomes? What is meant by two events are independent? How do you determine if two events are independent? How do you use the multiplication rule for independent events?

Sample Space The sample space S of random phenomenon is the set of all possible outcomes.

Event An event is any outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space

Sample Space: Flipping a Coin and Rolling a Die

Sample Space for two die

Sample Space: Rolling 2 Dice 1 2 3 4 5 6 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

Multiplication Principle If you can do one task n1 number of ways and a second task in n2 number of ways, then both tasks can be done in n1 x n2 number of ways.

Describe the Sample Space Problem 6.30 In each of the following situations, describe a sample space S for the random phenomenon. In some cases you have some freedom in specifying S, especially in setting the largest and the smallest value of S. a). Choose a student in your class at random. Ask how much time that student spent studying during the past 24 hours. Answer: S = { All numbers between 0 and 24} b). The Physicians’ Health Study asked 11,000 physicians to take an aspirin every other day and observed how many of them had heart attack in a five year period. Answer: S = { All integers from 0 to 11,000} c). In a test of a new package design, you drop a carton of a dozen eggs from a height of 1 foot and count the number of broken eggs. Answer: S = { Integers from 0 to 12}

Probability Model A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events.

Assigning Probability The probability of any event A is

Assigning Probability to Events Suppose we roll two die and we are interested in assigning a probability to total number of dots showing face up on the die. 1 2 3 4 5 6 7 8 9 10 11 12

Assigning Probability to Events in S 2 3 4 5 6 7 8 9 10 11 12 P(A) 1/36 1/18 1/12 1/9 5/36 1/6

Additional Problems Problem 6.36 Suppose you select a card from a standard deck of 52 playing cards. In how many ways can the selected card be – a). A red card? Ans: 26 P(red card) = 26/52 = 0.5 b). A heart? Ans: 13 P(A heart) = 13/52 = 0.25 c). A queen and a heart? Ans: 1 P(A queen and a heart) = 1/52 = 0.019 d). A queen or a heart? Ans: 16 P(A queen or a heart) = 16/52 = 0.308 e). A queen that is not a heart? Ans: 3 P(A queen that is not a heart) = 3/52 = 0.058

Probability Models Summary

Part 2 Probability Rules Facts that must be true for any assignment of probabilities. Facts follow the idea of probability as “the long-run proportion of repetitions on which an event occurs

Probability Rules: Rule 1 Any probability is a number between 0 and 1.

Probability Rules: Rule 2 The sum of the probabilities of all possible outcomes must equal 1.

Probability Rules: Rule 3 If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.

Probability Rules: Rule 3

Venn Diagram for Two Disjointed Events

Probability Rules: Rule 4 The probability that an event does not occur is 1 minus the probability that the event does occur.

Venn Diagram for the Complement Ac of Event A

Summary of Probability Rules

Example What is P(Married)? Marital status: Never married Married Widowed Divorced Probability: .298 .622 .005 .075 What is P(Married)? P(Married)=.622

Example What is P(Never married or Divorced)? Marital status: Widowed Divorced Probability: .298 .622 .005 .075 What is P(Never married or Divorced)? Since “Never married and Divorced are disjoint, P(Never married or Divorced)= .298+.075=.373 (Addition Rule for disjoint events)

Example What is P(not Married)? Marital status: Never married Married Widowed Divorced Probability: .298 .622 .005 .075 What is P(not Married)? P(not Married)= 1-.622=.378 (Complement Rule)

Benford’s Law Example 6.15 Benford’s Law is the distribution of first digits in tax records, payment records, invoices, etc. The next slide gives the distribution for legitimate records. Since crooks avoid using too many round number and fake data by using random digits, the illegitimate records will end up with too many first digits 6 or greater and too few 1s and 2s. This distribution is handy in spotting illegitimate records.

Example First Digit 1 2 3 4 5 6 7 8 9 Probability: .301 .176 .125 .097 .079 .067 .058 .051 .046

Example First Digit 1 2 3 4 5 6 7 8 9 Probability: .301 .176 .125 .097 .079 .067 .058 .051 .046

Example First Digit 1 2 3 4 5 6 7 8 9 Probability: .301 .176 .125 .097 .079 .067 .058 .051 .046

Special Case – Equally Likely Outcomes Some random phenomenon have balance which produces equally likely outcome, such as flipping coins and drawing playing cards. Most random phenomenon do not have equally likely outcomes, so the general rule is more important.

Venn Diagrams: Disjoint Events B Example: Flipping one coin. Heads or Tails can not occur at the same time for each event. “OR”

Venn Diagrams: Non-disjoint Events B A A and B Example: Flipping two coins at the same time. Two head or two tails can exist at the same time.

The Probability of Event A and Event B P(A and B) This rule applies only to independent events.

Are these events Independent? Event A: Randomly selected person is a man. Event B: Randomly selected person is pregnant.

Proof that independent events cannot be disjoint events Theorem – If A and B are both non-empty, independent events, then A and B can not be disjoint (i.e., they have to have outcomes in common). Proof ( by contradiction) Assume that A and B are non-empty, independent events. Since A and B are independent, then P(A and B) = P(A)●P(B). Suppose A and B are disjoint, Then P(A and B) = P( Null) = 0 Then P(A) = 0 or P(B) = 0. This means A and B are empty sets. But this contradicts our assumption that A and B are non-empty sets. Therefore, we conclude that A and B are not disjoint and do intersect.

One Last Problem Choose a person aged 19 to 25 years at random and ask, “In the past seven days, how many times did you go to an exercise or fitness center or work out?” Based on a large sample survey, here is a probability model for the answer you will get: Days 1 2 3 4 5 6 7 Probability 0.68 0.05 0.07 0.08 0.04 0.01 0.02 Is this a legitimate probability model? What is the probability that the person you choose worked out either 2 or 3 days in the past seven days? What is the probability that the person you choose worked out at least one day in the past seven days?