Todays Goals

1
Todays Goals

• Introduction to Probability
• Perform basic probability calculations
• Apply Bayes Theorem
• Homework 1 due Today
• Homework 2 (due Thursday September 25)
• GPCs New Product Decision
• For 1, just draw the ID
• For 2, use tools provided with book.
• Strenlar
• Screening for Colorectal Cancer (Ch. 7)

2
Example

• A pharmaceutical company is deciding whether to
  invest in RD for a new product or not. First
  they must decide whether to perform the first
  test. Once they have the results of the first
  test they can then decide whether to perform the
  second test. The first test has a cost of
  250,000, and a probability of success of 0.1. If
  the first test is not successful, the drug will
  fail for sure. The second test costs 1M, and has
  probability 50 of being successful. If the drug
  is successful it will earn 10M. If it is not
  successful, it will earn 0.
• Draw the decision tree for this problem, and roll
  it back to determine optimal strategy.

3
Decision Tree
10-.25-1 8.75M
Continue
P.1
-1-.25 -1.25M
Invest in first stage
-.25M
Stop
Continue
-1-.25M
1-P.9
-.25M
Stop
0
Dont Invest in first stage
4
Decision Tree
10-.25-1 8.75M
3.75
Continue
3.75
P.1
-1-.25 -1.25M
Invest in first stage
.15
-.25M
Stop
Continue
-1-.25M
1-P.9
-.25M
-.25
Stop
0
Dont Invest in first stage
5
Probability Basics

• Conditional Probability, independence
• Bayes Theorem
• Random Variables CDFs PDFs
• Moments
• Stochastic Dominance

6
Probability Measuring Uncertainty

• Elementary Event
• Sample space of all elementary events
• Events or Outcomes are subsets of the sample
  space
• We can use set theory to think about probability

7
S

• P(A) area(A)/area(S)
• P(A and B) area(AnB)/area(S)
• P(A or B) area (AuB)/area(S)
• P(A) P(B) p(AnB)

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S

• P(AB) area(AnB)/area(B)
• P(A and B)/P(B)

9
Independence

• chance events A and B are independent if and only
  if
• This implies that P(Ai and Bj) P(Ai )P(Bj )
• When chance events are independent, we can
  multiply their probabilities to get joint
  probabilities.

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Relevance

• If two chance events are not independent, then
  they are dependent.
• In DA we are trying to get away from this usage
  since it can be misleading.
• You will often read relevant in place of
  dependent

B
A
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Relevance and causal relationships

• Dependence/Relevance does not imply causality.
• Leading indicators and economy.
• Stork births and human births.

12
Conditional Independence
A
B
A and B are relevant.
13
Conditional Independence
A
B
C
A and B are relevant.
A and B are conditionally independent, given C.
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Conditional Independence
A
B
C
A and B are relevant.
A and B are conditionally independent, given C.
Events A and B are conditionally independent
given C iff
15
Complements and Total Probability

• The complement of outcome B is written
• This means that B did not occur.
• The total probability of an event occurring can
  be written
• This is often a very convenient way to assess the
  probability of an event.
• What is the probability that a stock goes up,
  given the dow jones goes up (down)?
• What is the probability that the dow jones goes
  up?

16
Bayes Theorem
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Bayes Theorem
P(B) is called the prior probability of B
P(AB), P(A-B) are called the liklihoods P(BA)
is called the posterior. This formula is very
useful when you are thinking of gathering
information about an event.
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Bayes Rule example

• A regulator is concerned a firm may be polluting
  the river. Let A firm pollutes. He thinks P(A)
  0.6
• He takes a sample of water and finds it polluted.
  Let this be event B.
• He thinks that P(BA) 0.9 and P(B-A) 0.3
• What is his posterior probability that they are
  polluting?

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• P(A) 0.6
• P(BA) 0.9
• P(B-A) 0.3

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What happens if P(BA) 1

• P(A) 0.6
• P(BA) 0.9
• P(B-A) 0.3

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What happens if P(B-A) 0

• P(A) 0.6
• P(BA) 0.9
• P(B-A) 0.3

22
Disease Tests Risky Business

• What probability statements can be inferred from
  the article?

23
Initial Assessment
Disease?
Test results
If the patient has the disease the test will
correctly report this 80 of the time. P(YD)
.8 If the patient does not have the disease, the
test will correctly report this 75 of the time.
P(NND) .75 The prior probability that the
patient has the disease is .1
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Flip the arc
P(YD) .8 P(NND) .75 P(D) .1
Disease?
Test results
What is the probability that the patient has the
disease given a positive test? Given a negative
test? How does this change if the prior
probability of the patient having the disease is
.01 or .5? Apply Bayes theorem to answer these
questions.
25
Random Variable

• A Random Variable is a Real-valued function over
  a sample space.
• Let x 1 if you get at least one head in 5 flips,
  0 otherwise.
• Sample space is HHHHH HHHHT HHHTH
• Let x the distance from Amherst to Atlanta, GA
  Sample space is the positive real numbers.

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Random Variable

• A random variable Y is said to be discrete if it
  can assume only a finite or countably finite
  number of distinct values.
• Discrete Let x 1 if you get at least one head
  in 5 flips, 0 otherwise.
• Sample space is HHHHH HHHHT HHHTH
• Continuous Let x the distance from Amherst to
  Atlanta, GA
• Sample space is the positive real numbers.

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Probability distribution of a discret RV

• The probability that Y takes on the value of y,
  P(Y y), is defined as the sum of the
  probabilities of all sample points in S that are
  assigned the value y
• Notation P(Y y) p(y)
• The probability distribution for a discrete
  random variable Y can be represented by a
  formula, a table, or a graph, which provides the
  probabilities p(y) corresponding to each and
  every value of y.

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Probability Mass Function
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Cumulative Distribution Function

• The CDF gives the probability that a random
  variable is less than or equal to a value.
• The CDF is continuous even when the RV is
  discrete.

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Cumulative Distribution Function
P(Yy)
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Continuous Distributions

• The uncertain quantity can take on any value
  within a range.
• The probability of equaling any particular value
  is zero, since there are infinitely many values
  it can equal.
• We can still construct a CDF P(Yy) F(y)

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Probability Distribution Function

• Analogous to the PMF is a Probability Density
  Function, f(.)

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Example

• Let f(x) 2-2x for x between 0 and 1.
• Is this a pdf?
• What is P(X ½)?
• What is P(X lt ½)?
• What is P(X ½)?
• What is F(x)?

34
Bayes Rule for continuous distributions

• L(yx) is the likelihood function for getting
  signal y given that the value of Xx
• p(x) prior probability density of X
• P(xy) is the posterior probability density over
  X, after getting signal y.

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Moments

• Expected Value (first moment)
• Variance and Standard Deviation (based on the 2nd
  moment)
• Covariance and Correlation

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Expected Value
Only if g is linear is it true that
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The Flaw of Averages

• Jensens inequality
• If g is convex then
• What if g is concave?
• What if g is linear?
• What if g is neither convex nor concave?

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The Flaw of Averages

• Let X 1, 2, 3 with equal likelihood
• What is E1/x?

39
A sobering example of the Flaw of Averages (taken
from Sam Savages Insight.xla)
40
Flaw of Averages Example

• You are managing a project.
• There are 8 simultaneous tasks that need to get
  done. The project is done when all the tasks are
  done.
• The mean and median time for each task to get
  done is 3 months.

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Flaw of Averages Example

• You are managing a project.
• There are 8 simultaneous tasks that need to get
  done. The project is done when all the tasks are
  done.
• The mean and median time for each task to get
  done is 3 months.
• How long will it take you to finish the project,
  on average?

42
Flaw of Averages Example

• You are managing a project.
• There are 8 simultaneous tasks that need to get
  done. The project is done when all the tasks are
  done.
• The mean and median time for each task to get
  done is 3 months.
• What is the probability that you will finish the
  job in about 3 months?

43
Flaw of averages Example

• See simulation

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Variance and Standard Deviation

• var(X) EX-EX EX2 EX2
• SD(X)
• Var(aXb) a2 Var(X)
• Co-Variance E(X-EX)(Y-EY)
  EXY-EXEY
• Correlation rxy

45
Correlation and Independence

• If two variables have zero correlation, are they
  independent?
• If two variables are independent, do they have
  zero correlation?

46
Correlation

• What is correlation?
• Does correlation imply relevance/dependence?
• Does lack of correlation imply independence?

y
x
Are x and y correlated? Are they independent?
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Example

• Find the correlation between X and Y.
• Are they independent? Prove it.

48
Variance and Risk

• Variance is a measure of risk
• but it is not the only thing that matters.
• Deal A earn 1,000,000 with probability 0.1 or
  0 with probability 0.9
• Deal B earn 200,000 with probability 0.9 or
  lose 800,000 with probability 0.1
• Which would you prefer? What is the expected
  value and variance of A and of B?

49

• Roll back the tree to see whether the firms
  should refuse or accept the 3B offer.
• Now, draw the CDFs for the two law firms.
• Which firm is better?
• Will it depend on risk aversion?