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Lec4 1. April 18, 2005

Relations in categorical data

Conditional probability

Outline:

Conditional probability;

Independence of events; The Bayes rule;

Tree diagram;

Simpsons paradox.

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Conditional Probability

IfP(B) = 0, the conditional probability of

event A given B has occurred, denoted byP(A |B), is defined by,

P(A |B) =P(A and B)

P(B)

We can rewrite the formula:

P(A and B) = P(A |B)P(B)

= P(B |A)P(A)

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Example

A focus group of 10 consumers has been

selected to view a new TV commercial. After

the viewing, 2 members of the focus groupwill be randomly selected and asked to

answer detailed questions about the

commercial. The group contains 4 men and 6

women.

P(first chosen person is female) =?

P(second person is female | first person

is female) = ?

P(both people are female) =?

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Independence

Events A and B are independent if

P(A |B) = P(A)

or equivalently

P(B |A) = P(B)

P(A and B) = P(A)P(B)

This means that the probability that Aoccurs is unchanged by information aboutwhether B has occurred (and vice versa).

IfA1, . . . , An are independent,P(A1 and A2 and An)

= P(A1) P(An)

Note: When A and B are disjoint

P(A or B) = P(A) + P(B)

P(A and B) = P() = 0

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Example

John and Paul go duck hunting together.

Suppose that John hits a target with

probability 0.3 and Paul, independently,

with probability 0.1. They both fire one

shot at a duck.

Given that exactly one shot hits the

duck, what is the conditional

probability that it is Johns shot?

That it is Pauls?

Given that the duck is hit, what is the

conditional probability that John hitit? that Paul hit it?

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Example

A blood test is 99% effective in detecting

a certain disease when the disease is

present. However, the test also yields afalse-positive result for 2% of healthy

patients tested. Suppose 0.1% of the

population has the disease. What is the

probability that a randomly tested

individual actually has the disease given

that his or her test result is positive?

D = an individual has the disease

T = test result is positive

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The Tree Diagram

5% of male high school athletes go on to playat college level. Of these, 1.7% enter major

league professional sports. Given that he

doesnt compete in college, the probability

that a high school athelete reaches the

professional play is 0.01%.

A = {competes in college}

B = {competes professionally}

What is the probability that a high

school athlete competes in college and

enter major league professional sports?

What is the probability that a high

school athlete will go on to professional

sports?

What proportion of professional athletes

competed in college?

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Relationships in Categorical Data

A two-way table is a way to display the

data from two categorical variables.

Example: Surgery death rates in two

hospitals

Hospital A Hospital B

died 63 16

survived 2037 784total 2100 800

Hospital A: 3% of surgery patients die

Hospital B : 2% of surgery patients dieQuestion: Which hospital is safer?

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Example (cont.)

Patients are classified as being in eitherpoor or good conditions beforesurgery.

good cond. poor cond.

A B A Bdied 6 8 57 8

survived 594 592 1443 192

total 600 600 1500 200

For patients in good condition:

hospital A: 1% of surgery patients die

hospital B: 1.3% of surgery patients die

For patients in poor condition:

hospital A: 3.8% of surgery patients die

hospital B: 4% of surgery patients die

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An association or comparison that holdsfor all of several groups can reversedirection when the data are combined toform a single group. This reversal is called

Simpsons paradox.

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