Conditional Probabilities Statistical Independence. Week 6 (1)

Сентябрь 14, 2022

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Conditional Probabilities Statistical Independence. Week 6 (1)

Содержание

2. Interpreting probability No matter what method is used to assign probabilities, we interpret the probability, using
5. Definition of marginal probability Represent the totals found in the margins of a contingency table: Marginal
6. Two events occur together : Joint probabilities
7. The following contingency table shows opinion about global warming among U.S. adults, broken down by political
8. A) What is the probability that a U.S. adult selected at random believes that global warming
9. DR SUSANNE HANSEN SARAL
10. Conditional Probability What if we were given the information that the selected person is a republican,
11. Conditional Probability
12. Conditional Probability When we restrict our focus to republicans, we look only at the republican’s row
13. Conditional Probability
14. Conditional Probability
15. Conditional Probability
16. Conditional Probability
17. Conditional Probability A conditional probability is the probability of one event, given that another event has
18. Conditional Probability - Example What is the probability that a car has a CD player, given
19. Conditional Probability Example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM No CD CD Total AC .2 .5 .7
20. Conditional Probability Example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM No CD CD Total AC .2 .5 .7
21. Conditional Probability – manufacturing example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM
22. Conditional Probability - example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM
23. Conditional Probability - example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM
24. Conditional Probability - example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM
25. Class exercises Discrimination in the workplace is illegal and companies that do so are often sued.
26. Class exercises What is the probability of promotion given female assistant professors ? What is the
27. Class exercises (continued) DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM
28. Statistical Independence Two events are statistically independent if and only if: Events A and B are
29. Statistical independence
30. Statistical Independence Car example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM No CD CD Total AC .2 .5
31. Class exercise Independent-dependent events Determine whether the event A3 and B2 are independent or dependent events.
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Interpreting probability
No matter what method is used to assign probabilities, we

interpret the probability, using the relative frequency approach for an infinite number of trails.
The probability is only an estimate (Turkish: tahmin), because the relative frequency approach defines probability as the “long-run” relative frequency.
The larger the number of observations the better the estimate will become.
Ex.: Tossing a coin
Head and tail will only occur 50 % in the long run
Computer simulations

DR SUSANNE HANSEN SARAL

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Definition of marginal probability
Represent the totals found in the margins

of a contingency table:
Marginal probabilities

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Two events occur together : Joint probabilities

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The following contingency table shows opinion about global warming among

U.S. adults, broken down by political party affiliation.

DR SUSANNE HANSEN SARAL

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A) What is the probability that a U.S. adult selected

at random believes that global warming is a serious problem?
B) What type of probability did you find in part A? (marginal or joint probability)
C) What is the probability that a U.S. adult selected at random is a Republican and believes that global warming is a serious issue?
D) What type of probability did you find in part C? (marginal or joint probability)

DR SUSANNE HANSEN SARAL

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DR SUSANNE HANSEN SARAL

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Conditional Probability
What if we were given the information that the

selected person is a republican, instead of U.S. adult? Would that change the probability that the selected person’s opinion about global warming is a nonissue?

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Conditional Probability

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Conditional Probability
When we restrict our focus to republicans, we look

only at the republican’s row of the table,
Which gives the conditional probability of opinion on Global warming “given” the person is republican.

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Conditional Probability

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Conditional Probability

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Conditional Probability

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Conditional Probability

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Conditional Probability
A conditional probability is the probability of one event, given

that another event has already occurred:

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The conditional probability of A given that B has occurred

The conditional probability of B given that A has occurred

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Conditional Probability - Example
What is the probability that a car has

a CD player, given that it has AC ?
i.e., we want to find P(CD | AC)

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

Of the cars in a second hand car dealer’s shop, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

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Conditional Probability Example
DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM
No CD
CD
Total
AC
.2
.5
.7
No AC
.2
.1
.3
Total
.4
.6
1.0
Of the

cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD).
20% of the cars have both.

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Conditional Probability Example
DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM
No CD
CD
Total
AC
.2
.5
.7
No AC
.2
.1
.3
Total
.4
.6
1.0
Given AC, we

only consider the top row (70% of the cars). Of these, 20% have a CD player.

(continued)

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Conditional Probability – manufacturing example

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

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Conditional Probability - example

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

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Conditional Probability - example

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

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Conditional Probability - example

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

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Class exercises
Discrimination in the workplace is illegal and companies that do

so are often sued. The female professors at a large university recently sued the university, about the last round of promotions from assistant professor to associate professor:
An analysis of the relationship between gender and promotion produced the following probabilities:

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

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Class exercises
What is the probability of promotion given female assistant professors

?
What is the probability of promotion given male assistant professors?
Is it reasonable to accuse the university of gender difference?

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

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Class exercises (continued)

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

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Statistical Independence
Two events are statistically independent if and only if:
Events

A and B are independent when the probability of one event is not affected by the other event
If A and B are independent, then

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

if P(B)>0

if P(A)>0

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Statistical independence

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Statistical Independence Car example
DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM
No CD
CD
Total
AC
.2
.5
.7
No AC
.2
.1
.3
Total
.4
.6
1.0
(continued)

So

the two events are not statistically independent

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Class exercise Independent-dependent events
Determine whether the event A3 and B2

are independent or dependent events. P(A3|B2) = P(A3)?
Determine whether the events B1 and A1 are independent or dependent events. P(A1|B1) =P(A1)?

DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

Conditional Probabilities Statistical Independence. Week 6 (1)

Содержание

Interpreting probabilityNo matter what method is used to assign probabilities, we

Definition of marginal probabilityRepresent the totals found in the margins

Two events occur together : Joint probabilities

The following contingency table shows opinion about global warming among

A) What is the probability that a U.S. adult selected

DR SUSANNE HANSEN SARAL

Conditional ProbabilityWhat if we were given the information that the

Conditional Probability

Conditional ProbabilityWhen we restrict our focus to republicans, we look

Conditional Probability

Conditional Probability

Conditional Probability

Conditional Probability

Conditional ProbabilityA conditional probability is the probability of one event, given

Conditional Probability - ExampleWhat is the probability that a car has

Conditional Probability ExampleDR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COMNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.0Of the

Conditional Probability ExampleDR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COMNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.0Given AC, we

Conditional Probability – manufacturing example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

Conditional Probability - example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

Conditional Probability - example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

Conditional Probability - example DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

Class exercisesDiscrimination in the workplace is illegal and companies that do

Class exercisesWhat is the probability of promotion given female assistant professors

Class exercises (continued) DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

Statistical IndependenceTwo events are statistically independent if and only if:Events

Statistical independence

Statistical Independence Car exampleDR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COMNo CDCDTotalAC.2.5.7No AC.2.1.3Total.4.61.0(continued) So

Class exercise Independent-dependent events Determine whether the event A3 and B2

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When we restrict our focus to republicans, we look

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A conditional probability is the probability of one event, given

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Discrimination in the workplace is illegal and companies that do

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What is the probability of promotion given female assistant professors

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DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM

Statistical Independence
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DR SUSANNE HANSEN SARAL, SUSANNEHANSENSARAL@GMAIL.COM
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