Transcript Lecture 4 - Statistics

Statistics 400 - Lecture 4
 Today - 4.1-4.5
 Suggested Problems: 2.1, 2.48 (also compute mean), construct
histogram of data in 2.48
Probability (Chapter 4)
 “There is a 75% chance of rain tomorrow”
 What does this mean?
Definitions
 Probability of an outcome is a numerical measure of the chance of
the outcome occurring
 A experiment is random if its outcome is uncertain
 Sample space, S, is the collection of possible outcomes of an
experiment
 Event is a set of outcomes
 Event occurs when one of its outcomes occurs
Example
 A coin is tossed 2 times
 S=
 Describe event of getting 1 heads and 1 tails
 Probability of an event is the long-term proportion of times the
event would occur if the experiment is repeated many times
 Probability of event, A is denoted P(A)
 0  P( A)  1
 P(A) is the sum of the probabilities for each outcomes in A
 P(S)=1
Discrete Uniform Distribution
 Sample space has k possible outcomes S={e1,e2,…,ek}
 Each outcome is equally likely
 P(ei)=
 If A is a collection of distinct outcomes from S, P(A)=
 Bag of balls has 5 red and 5 green balls
 3 are drawn at random
 S=
 A is the event that at least 2 green are chosen
 A=
 P(A)=
Example (pg 140)
 Inherited characteristics are transmitted from one generation to the
next by genes
 Genes occur in pairs and offspring receive one from each parent
 Experiment was conducted to verify this idea
 Pure red flower crossed with a pure white flower gives
 Two of these hybrids are crossed. Outcomes:
 Probability of each outcome
 Sometimes, not all outcomes are equally likely (e.g., fixed die)
 Recall, probability of an event is long-term proportion of times the
event occurs when the experiment is performed repeatedly
 NOTE: Probability refers to experiments or processes, not
individuals
Probability Rules
 Have looked at computing probability for events
 How to compute probability for multiple events?
 Example: 65% of Umich Business School Professors read the Wall
Street Journal, 55% read the Ann Arbor News and 45% read both.
A randomly selected Professor is asked what newspaper they read.
What is the probability the Professor reads one of the 2 papers?
 Addition Rule:
P( A  B)  P( A)  P( B)  P( A  B)
 If two events are mutually exclusive:
P( A  B)  P( A)  P( B)
 Complement Rule
P( A)  1  P( A )
Conditional Probability
 Sometimes interested in in probability of an event, after information
regarding another event has been observed
 The conditional probability of an event A, given that it is known B
has occurred is:
P( A  B)
P( A | B) 
 Called “probability of A given B ”
P( B)
Example
 In a region 12% of adults are smokers, 0.8% are smokers with
emphysema and 0.2% are non-smokers with emphysema
 What is the probability that a randomly selected individual has
emphysema?
 Given that the person is a smoker, what is the probability that the
person has emphysema?
 Multiplication rule for conditional probability:
P( A  B)  P( A | B) P( B)
 Can use any 2 of the probabilities to get the third
Independent Events
 Two events are independent if:
P( A | B)  P( A)
 The intuitive meaning is that the outcome of event B does not
impact the probability of any outcome of event A
 Alternate form:
P( A and B)  P( A) P( B)
Example
 Flip a coin two times
 S=
 A={head observed on first toss}
 B={head observed on second toss}
 Are A and B independent?
Example
 Mendel used garden peas in experiments that showed inheritance
occurs randomly
 Seed color can be green or yellow
 {G,G}=Green otherwise pea is yellow
 Suppose each parent carries both the G and Y genes
 M ={Male contributes G}; F ={Female contributes G}
 Are M and F independent?
Example (Randomized Response Model)
 Can design survey using conditional probability to help get honest
answer for sensitive questions
 Want to estimate the probability someone cheats on taxes
 Questionnaire:
 1. Do you cheat on your taxes?
 2. Is the second hand on the clock between 12 and 3?

YES
NO
 Methodology: Sit alone, flip a coin and if the outcome is heads
answer question 1 otherwise answer question 2