Download Statistics 400 - Lecture 2

Statistics 270 - Lecture 6
• Last day: Probability rules
• Today: Conditional probability
• Suggested problems: Chapter 2: 45, 47, 59, 63, 65
Example – Let’s Make a Deal:
• A contestant is given a choice of three doors of which one
contained a prize such as a Car
• The other two doors contained gag gifts like a chicken or a donkey
• After the contestant choses an initial door, the host of the show
reveals an empty door among the two unchosen doors, and asks
the contestant if they would like to switch to the other unchosen
door
• What should the contestant do?
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)
P( B)
• Called “probability of A given B ”
Example – Let’s Make a Deal:
• A contestant is given a choice of three doors of which one
contained a prize such as a Car
• The other two doors contained gag gifts like a chicken or a donkey
• After the contestant choses an initial door, the host of the show
reveals an empty door among the two unchosen doors, and asks
the contestant if they would like to switch to the other unchosen
door
• What should the contestant do?
Some Useful Formulas
•
Multiplication Rule:
P( A  B)  P( A) P( B | A)  P( B) P( A | B)
•
Law of Total Probability:
P( A)  P( A | B) P( B)  P( A | B' ) P( B' )
•
Bayes Theorem:
P( B | A) P( A)
P( A | B) 
P( B | A) P( A)  P( B | A' ) P( A' )
Where do these come from?
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?
Example
•
From a group of 5 Democrats, 5 Republicans and 5 Independents,
a committee of size 3 is to selected
•
What is the probability that each group will be represented on the
committee if the first person selected is an Independent?
Example
•
Consider a routine diagnostic test for a rare disease
•
Suppose that 0.1% of the population has the disease, and that
when the disease is present the probability that the test indicates
the disease is present is 0.99
•
Further suppose that when the disease is not present, the
probability that the test indicates the disease is present is 0.10
•
For the people who test positive, what is the probability they
actually have the disease
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
•
Several Events
•
Suppose (A1, A2, …, Ak) form a partition of the sample
space…i.e., they are mutually exclusive and their union
equals the sample space
k
P( B)   P( B | Ai ) P( Ai )
i 1
•
Bayes Theorem: suppose (A1, A2, …, Ak) form a
partition of the sample space
P( A j | B) 
P( B | A j ) P( A j )
k
 P( B | A ) P( A )
i 1
i
i
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?
Several Independent Events
•
Events A1, A2, …, An are mutually independent if for every k (k=2, 3, …, n) and
every index set i1, …, ik
P( Ai1  Ai2  ...  Aik )  P( Ai1 ) P( Ai2 )...P( Aik )
•
That is, events are mutually independent if the probability of the intersection of any
subset of the n events is equal to the product of the individual probabilities