Probabilities. Week 5 (2)

Содержание

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Where do probabilities come from? Two different ways to determine probabilities:

Where do probabilities come from?

Two different ways to determine probabilities:
1.

Objective approach:
a. Relative frequency approach, derived from historical data
b. Classical or logical approach based on logical observations, ex. Tossing a
fair coin
2. Subjective approach, based on personal experience

DR SUSANNE HANSEN SARAL

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Types of Probability Relative frequency approach Objective Approach: a) Relative frequency

Types of Probability Relative frequency approach

Objective Approach:
a) Relative frequency
We calculate

the relative frequency (percent) of the event:

2 –

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Objective probability – The Relative Frequency DR SUSANNE HANSEN SARAL Hospital

Objective probability – The Relative Frequency

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Hospital Unit Number of Patients Relative Frequency
Cardiac Care 1,052 11.93 %
Emergency 2,245 25.46 %
Intensive Care 34 3.86 %
Maternity 552 6.26 %
Surgery 4,630 52.50 %
Total: 8,819 100.00 %

P (cardiac care) =

 

Total number of patient admitted to the hospital

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Objective probability assessment – The Relative Frequency Approach DR SUSANNE HANSEN

Objective probability assessment – The Relative Frequency Approach

DR SUSANNE

HANSEN SARAL

Example: Hospital Patients by Unit per semester
Hospital Unit Number of Patients Relative Frequency
Cardiac Care 1,052 11.93 %
Emergency 2,245 25.46 %
Intensive Care 340 3.86 %
Maternity 552 6.26 %
Surgery 4,630 52.50 %
Total: 8,819 100.00 %
The 2 probability rules are satisfied:
Individual probabilities are all between 0 and 1
0 ≤ P (event) ≤ 1
Total of all event probabilities equals 1
∑ P (event) = 1.00

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Types of Probability Classical approach Objective Approach: DR SUSANNE HANSEN SARAL

Types of Probability Classical approach

Objective Approach:

DR SUSANNE HANSEN SARAL

b)

Classical approach:

♥ ♣ ♦ ♠

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Subjective approach to assign probabilities We use the subjective approach :

Subjective approach to assign probabilities
We use the subjective approach :
No

possibility to use the classical approach nor the relative frequency approach.
No historic data available
New situation that nobody has been in so far
The probability will differ between two people, because it is subjective.

DR SUSANNE HANSEN SARAL

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Types of Probability Subjective Approach: Based on the experience and judgment

Types of Probability
Subjective Approach:
Based on the experience and judgment of the

person making the estimate:
Opinion polls (broad public)
Judgement of experts (professional judgement)
Personal judgement

DR SUSANNE HANSEN SARAL

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Interpreting probability No matter what method is used to assign probabilities,

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 experiments.
The probability is only an estimate, 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, birth of a baby, etc.
Head and tail will only occur 50 % in the long run
Girl and boy will only occur 50 % in the long run

DR SUSANNE HANSEN SARAL

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Probability rules continued Rule 1 and 2 If A is any

Probability rules continued Rule 1 and 2

If A is any event

in the sample space S, then
a probability is a number between 0 and 1
The probability of the set of all possible outcomes must be 1
P(S) = 1 P(S) = Σ P(Oi ) = 1 , where S is the sample space

DR SUSANNE HANSEN SARAL

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Objective probability assessment – The Relative Frequency Approach DR SUSANNE HANSEN

Objective probability assessment – The Relative Frequency Approach

DR SUSANNE

HANSEN SARAL

Example: Hospital Patients by Unit per semester
Hospital Unit Number of Patients Relative Frequency
Cardiac Care 1,052 11.93 %
Emergency 2,245 25.46 %
Intensive Care 340 3.86 %
Maternity 552 6.26 %
Surgery 4,630 52.50 %
Total: 8,819 100.00 %
Individual probabilities are all between 0 and 1
0 ≤ P (event) ≤ 1
Total of all event probabilities equals, S
P(s) = ∑ P (event, O) = 1.00

Слайд 12

Probability rules. Rule 3 Complement rule Suppose the probability that you

Probability rules. Rule 3 Complement rule
Suppose the probability that you win

in the lottery is 0.1 or 10 %.
What is the probability then that you don’t win in the lottery?

DR SUSANNE HANSEN SARAL

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Probability rules. Rule 3 Complement rule DR SUSANNE HANSEN SARAL

Probability rules. Rule 3 Complement rule

 

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Probability rule 4 Multiplication rule – calculating joint probabilities Independent events DR SUSANNE HANSEN SARAL

Probability rule 4 Multiplication rule – calculating joint probabilities Independent

events

 

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Слайд 15

Multiplication Rule for independent events (continued) DR SUSANNE HANSEN SARAL

Multiplication Rule for independent events (continued)

 

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Independent events Events are independent from each other when the probability

Independent events
Events are independent from each other when the

probability of occurrence
of the first event does not affect the probability of occurrence of the second
event.
The probability of occurrence of the second event will be the same as for the
first event.

DR SUSANNE HANSEN SARAL

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Multiplication rule – calculating joint probabilities Dependent events DR SUSANNE HANSEN SARAL

Multiplication rule – calculating joint probabilities Dependent events

 

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Multiplication rule – Dependent events (continued) DR SUSANNE HANSEN SARAL

Multiplication rule – Dependent events (continued)

 

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Multiplication Rule - Dependent events (continued) DR SUSANNE HANSEN SARAL

Multiplication Rule - Dependent events (continued)

 

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Multiple choice quiz: 1 correct 3 false You are going to

Multiple choice quiz: 1 correct 3 false

You are going

to take a multiple choice exam. You did not have time to study and will
therefore guess. The questions are independent from each other.
There are 5 multiples choice questions with 4 alternative answers. Only one answer
is correct.
What is the probability that you will pick the right answer out of the 4 alternatives?
What is the probability that you will pick the wrong answer out of the 4 alternatives?
What is the probability that you will pick two answers correctly? What is the probability of
picking two wrong answers? What is the probability that you will pick all the correct answers out
of the 5 questions? What is the probability that you will pick all wrong answers out of the 5
questions?

DR SUSANNE HANSEN SARAL

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Multiple choice quiz: 1 correct 3 false DR SUSANNE HANSEN SARAL

Multiple choice quiz: 1 correct 3 false

 

DR SUSANNE HANSEN SARAL

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Probability Rule 5: Addition rule for mutually exclusive events DR SUSANNE HANSEN SARAL

Probability Rule 5: Addition rule for mutually exclusive events

 

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HANSEN SARAL
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Probability rule 5: Addition rule for mutually exclusive events Example DR SUSANNE HANSEN SARAL

Probability rule 5: Addition rule for mutually exclusive events Example

 

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Addition rule of mutually exclusive events: Example – Definition of events DR SUSANNE HANSEN SARAL

Addition rule of mutually exclusive events: Example – Definition of events

 

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SUSANNE HANSEN SARAL
Слайд 25

Addition rule of mutually exclusive events: Example - Solution DR SUSANNE HANSEN SARAL

Addition rule of mutually exclusive events: Example - Solution

 

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SUSANNE HANSEN SARAL
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Addition rule of mutually exclusive events: Class exercise A corporation receives

Addition rule of mutually exclusive events: Class exercise

A corporation receives

a shipment of 100 units of computer chips from a manufacturer.
Research indicates the probabilities of defective parts per shipment shown in the following table:
What is the probability that there will be fewer than three defective parts in a shipment? P(x < 3)
What is the probability that there will be more than one defective part in a shipment? P(x > 1)
The five probabilities in the table sum up to 1. Why must this be so?

DR SUSANNE HANSEN SARAL

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Probability rule 6: Addition rule for non- mutually exclusive events A∩B

Probability rule 6: Addition rule for non- mutually exclusive events


A∩B

A

B

S

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Слайд 28

DR SUSANNE HANSEN SARAL Probability rule 6: Addition rule for non-mutually exclusive events


DR SUSANNE HANSEN SARAL

Probability rule 6: Addition rule for non-mutually

exclusive events
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Addition rule of mutually non-exclusive events rolling a dice DR SUSANNE

Addition rule of mutually non-exclusive events rolling a dice

DR SUSANNE HANSEN

SARAL

Ch. 3-

S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]

Слайд 30

Addition rule of mutually non-exclusive events: Example: P (A U B)

Addition rule of mutually non-exclusive events: Example: P (A U

B) = P(A) + P(B) – P(A ∩ B)

A video store owner finds that 30 % of the customers entering the store ask an assistant for help, and that 20 % of the customers buy a video before leaving the store.
It is also found that 15 % of all customers both ask for assistance and make a purchase.
What is the probability that a customer does at least one of these two things?

DR SUSANNE HANSEN SARAL

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Addition rule of non-mutually exclusive events: Example: A video store owner

Addition rule of non-mutually exclusive events: Example:

A video store owner

finds that 30 % of the customers entering the store ask an assistant for help, and that 20 % of the customers buy a video before leaving the store. It is also found that 15 % of all customers both ask for assistance and make a purchase.
What is the probability that a customer does at least one of these two things?

DR SUSANNE HANSEN SARAL

Слайд 32

Addition rule of non-mutually exclusive events: P(A U B) = P(A)

Addition rule of non-mutually exclusive events: P(A U B) =

P(A) + P(B) – P(A ∩ B) Class exercise
It was estimated that 30 % of all students in their 4th year at a university campus were concerned about employment future. 25 % were seriously concerned about grades, and 20 % were seriously concerned about both.
What is the probability that a randomly chosen 4th year student from this campus is seriously concerned with at least one of these two concerns?

DR SUSANNE HANSEN SARAL

Слайд 33

Class exercise - solution DR SUSANNE HANSEN SARAL

Class exercise - solution

 

DR SUSANNE HANSEN SARAL

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Calculating probabilities of complex events Now we will look at how

Calculating probabilities of complex events
Now we will look at how to

calculate the probability of more complex
events from the probability of related events.
Example:
Probability of tossing a 3 with two dices is 2/36.
This probability is derived by combining two possible events:
tossing a 1 (1/36) and tossing a 2 (1/36)

DR SUSANNE HANSEN SARAL

Слайд 35

How to calculate probabilities of intersecting events DR SUSANNE HANSEN SARAL

How to calculate probabilities of intersecting events

 

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Слайд 36

Drawing a Card – not mutually exclusive Draw one card from

Drawing a Card – not mutually exclusive
Draw one card from a

deck of 52 playing cards
A = event that a 7 is drawn
B = event that a heart is drawn

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P (a 7 is drawn) = P(A)= 4/52 = 1/13
P (a heart is drawn) = P(B) = 13/52 = 1/4

These two events are not mutually exclusive since a 7 of hearts can be drawn
These two events are not collectively exhaustive since there are other cards in the deck besides 7s and hearts

Слайд 37

Joint probabilities - A business application A manufacturer of computer hardware

Joint probabilities - A business application
A manufacturer of computer hardware

buys microprocessors chips to use in the assembly process from two different manufacturers A and B.
Concern has been expressed from the assembly department about the reliability of the supplies from the different manufacturers, and a rigorous examination of last month’s supplies has recently been completed with the results shown:

DR SUSANNE HANSEN SARAL

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Manufacturer of computer hardware- Contingency table - joint probabilities DR SUSANNE HANSEN SARAL

Manufacturer of computer hardware- Contingency table - joint probabilities

DR SUSANNE

HANSEN SARAL
Слайд 39

Manufacturer of computer hardware Contingency table joint probabilities It looks as

Manufacturer of computer hardware Contingency table joint probabilities
It looks

as if the assembly department is correct in expressing concern. Manufacturer B is supplying a smaller quantity of chips in total but more are found to be defective compared with Manufacturer A.
However, let us consider this in the context of the probability principles we have developed:
Relative frequency method (based on available data)

DR SUSANNE HANSEN SARAL

Слайд 40

Manufacturer of computer hardware Marginal probabilities Let us consider the total

Manufacturer of computer hardware Marginal probabilities

Let us consider the total

of 9897 as a sample. Suppose we had chosen one chip at random from this sample. The following events and their probabilities can then be obtained:
Find the probability of the following – marginal probabilities :
Event A: the chip was supplied by Manufacturer A
Event B: the chip was supplied by Manufacturer B
Event C: the chip was satisfactory
Event D: the chip was defective

DR SUSANNE HANSEN SARAL

Слайд 41

Manufacturer of computer hardware Joint probabilities (Continued) Let us consider the

Manufacturer of computer hardware Joint probabilities (Continued)

Let us consider the

total of 9897 as a sample. Suppose we had chosen one chip at random from this sample. The following joint events and their probabilities can be obtained:
And the joint probabilities:
P(A and C) supplied by A and satisfactory Joint probabilities
P(B and C) supplied by B and satisfactory
P(A and D) Supplied by A and defective
P(B and D) supplied by B and defective

DR SUSANNE HANSEN SARAL

Слайд 42

Interpretation of the joint probabilities in the example The joint probability

Interpretation of the joint probabilities in the example

The joint probability

that a chip is defective and that it is delivered from Manufacturer A is 0.012
The joint probability that a chip is satisfactory and it is delivered by Manufacturer A is 0.589
The probability that a chip is satisfactory and it is delivered by Manufacturer B is 0.379
The probability that a chip is defective and it is delivered by Manufacturer B is 0.020

DR SUSANNE HANSEN SARAL

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Notations for the marginal and joint events DR SUSANNE HANSEN SARAL

Notations for the marginal and joint events

 

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Слайд 44

Marginal probabilities DR SUSANNE HANSEN SARAL

Marginal probabilities

 

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The following contingency table shows opinion about global warming among U.S.


The following contingency table shows opinion about global warming among

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

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Слайд 46

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


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?

DR SUSANNE HANSEN SARAL

Слайд 47

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


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

at random believes that global warming is a serious problem? 63 %
B) What type of probability did you find in part A? (marginal or joint probability) Marginal 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? 18 %
D) What type of probability did you find in part C? Joint probability

DR SUSANNE HANSEN SARAL

- Предыдущая
Conditional Probabilities Statistical Independence. Week 6 (1)
Следующая -
Empirical rule - Probabilities. Week 5 (1)

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