Download Lecture 2 - CMU Statistics

36-410: Introduction to Probability Modeling
Spring 2016
Lecture 2: Probability Review
Lecturer: Sivaraman Balakrishnan
2.1
Review
1. Announcements
2. Probability review: Sample space, events and probability measure
2.2
Outline
1. Basic rules to calculate probabilities and examples that might be helpful
for your assignment.
2.3
Some basic rules
Always keep in mind a Venn diagram.
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Lecture 2: Probability Review
2.3.1
2-2
The addition rule
What is P(A ∪ B)?
2.3.2
The complement rule
What is P(Ac )?
2.3.3
The ELO rule
This is not really a rule. IF every outcome in S is equally likely then what
can we say about P(A)?
Lecture 2: Probability Review
2-3
Applying this rule requires some thought on how to partition S into equally
likely outcomes.
Example 1:
Lecture 2: Probability Review
Example 2: A drug has the following information:
1. There is a 10% chance of experiencing headache (H).
2. There is a 15% chance of experiencing nausea (N).
3. There is a 5% chance of experiencing both side effects.
What is the probability of experiencing at least one side effect?
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Lecture 2: Probability Review
2.4
2-5
Counting problems
Several probability calculations involve counting combinations. The basic
rule is: if there are n distinct objects there are
n
n!
,
=
k!(n − k)!
k
possible combinations of size k.
Example 3: I give you 5 cards from a deck. What is the probability that
you get exactly 1 jack?
Lecture 2: Probability Review
Example 4:
house?
2-6
Poker hands: Which is more likely a four of a kind or a full
Lecture 2: Probability Review
2.5
2-7
Conditional Probability
Often we want to calculate the probability that an event A occurs given that
an event B occurred.
We use the notation: P(A|B).
This is only defined when P(B) > 0. The rule for conditional probability:
P(A|B) =
The Venn Diagram viewpoint:
Lecture 2: Probability Review
Example 5: Roll a die once:
A = {even number},
B = {1, 2, 3, 5}.
What is P(A|B)?
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Lecture 2: Probability Review
2-9
Example 6: I flip two coins, hidden from your view. You ask, “Is at least
one coin heads? ” I say, “Yes.” What is probability (from your perspective)
that both are heads?
Lecture 2: Probability Review
2.5.1
The Multiplication rule
I call this the chain rule.
P(A ∩ B) =
2.5.2
The Law of Total Probability
P(A) =
The Venn Diagram viewpoint:
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Lecture 2: Probability Review
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Example 7: A bank is considering extending credit to a new customer
and is interested in the probability that the client will default on the loan.
Based on historical data, the bank knows that there is a 5% chance that
a customer who has overdrawn an account will default, while there is only
a 0.5% chance that a customer who has never overdrawn an account will
default. Unfortunately, the bank does not know for sure if the customer will
overdraw her account. Based on background checks the bank believes there
is a 30% chance that the customer will overdraw the account. Calculate the
probability that she will default if credit is extended.
Lecture 2: Probability Review
2.5.3
Generalizing the Law of Total Probability
We say that B1 , B2 , . . . , Bn is a partition of S if
•
•
Given a partition of S we have that,
P(A) =
How does this generalize the previous version?
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Lecture 2: Probability Review
2.5.4
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Bayes’ rule
This rule is quite central to probability theory (and also statistics, machine
learning, etc.). First the simpler version:
P(B|A) =
You should think about using Bayes’ rule when you want to calculate P(B|A)
and P(A|B) is known or easier to calculate.
The general version: Suppose B1 , B2 , . . . Bn are a partition of S, and A is any
event
P(Bi |A) =
Lecture 2: Probability Review
2-14
Example 8: Polygraph tests If a person is lying, the probability that
this is correctly detected by the polygraph is 0.88, whereas if the person is
telling the truth, this is correctly detected with probability 0.86. Suppose we
are consider a question for which 99% of all subjects tell the truth.
Our polygraph machine says a subject is lying on this question. What is the
probability that the polygraph is incorrect?
Lecture 2: Probability Review
2-15
Example 9: There are 3 drawers, one full of white socks, the second full
of black socks, and the third half black and half white. Select one drawer
at random, set it aside. Then draw one sock randomly from each of the
remaining two drawers. What is the probability of getting a matching pair?
Lecture 2: Probability Review
2-16
Example 10: I have a stick of length 1. I place two marks uniformly at
random over the length of the stick and then break the stick at the two point.
I now have three pieces. What is the probability that these can be made into
a triangle?
Lecture 2: Probability Review
2.6
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Independence
One of the key concepts of this course: recall, Markov chains deviate from
the traditional assumption of independence.
Events A1 , A2 , . . . An are independent if
P(Ai1 ∩ Ai2 ∩ . . . ∩ Aik ) =
for every collection of distinct indices {i1 , i2 , . . . , ik }.
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Example 11: Consider a sequence of n independent trials, each of which
has a probability 1/n of being a “success”. What is the probability of zero
successes in n trials? What if the number of trials is doubled?