Basic Concepts in Probability

7/30/2019 Basic Concepts in Probability

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Basic Probability Concepts

(Part 1)


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Probability

Degree of certainty or uncertainty that a

certain event will occur

Ranges from 0 to 1

l---------------------------l----------------------------l0 0.5 1

Impossible Sure

Event Event


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STATISTICAL EXPERIMENT, SAMPLESPACE, AND EVENTS


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Experiment

A process that generates well-defined

experimental outcomes

experimental outcome = sample point

For a single repetition of an experiment, only

one of all possible outcomes occurs


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Experiment Experimental Outcomes

Tossing a fair coin Head, Tail

Rolling a fair die 1, 2, 3, 4, 5, 6

Playing a board game Win, Tie, Lose

Coming to class Early, On time, Late


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Experiment Experimental Outcomes

Tossing a fair coin H, T

Rolling a fair die 1, 2, 3, 4, 5, 6

Playing a board game W, T, L

Coming to class E, O, L


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Sample space

The set of all experimental outcomes / sample

points


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Experiment Sample Space

Tossing a fair coin S = {H, T}

Rolling a fair die S = {1, 2, 3, 4, 5, 6}

Playing a board game S = {W, T, L}

Coming to class S = {E, O, L}


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But there is such a thing as Multiple-Step experiments. . .

Tossing a fair coin twice

Rolling a fair die twice

Playing a board game thrice

Coming to class 26 times


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Counting Rule for Multiple-Step

Experiments

If an experiment can be described as a

sequence ofksteps with n1 possible outcomes

on the first step, n2 possible outcomes on the

second step, and so on, then the total number

of experimental outcomes is given by

(n1)(n2)...(nk)

(Anderson, Sweeney, & Williams, 2008).


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Multiple-Step

Experiment

Sample Space Total Number

of

Experimental

Outcomes

Tossing a faircoin twice

S = {(H,H), (T,H), (H,T), (T,T)} 2 x 2 = 4

Rolling a fair die

twice

S = {(1,1), (1,2), (1,3), ... ,(6,6)} 6 x 6 = 36

Playing a boardgame thrice

S = {(W,W,W), (W,W,T), (W,W,L),(W,T,W), ... , (L,L,L)}

3 x 3 x 3 = 27

Coming to class

26 times

S = {(E,E,...,E), ... , (L,L,...,L)} 3^26 =

2.54 x 10^12


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Consider the multiple-step experiment of (1) tossing a fair coin

then (2) playing a board game

Sample space of (1) S = {H, T}

Sample space of (2) S = {W, T, L}

Therefore, the sample space of this multiple-step

experiment is...

S = {(H,W), (H,T), (H,L), (T,W), (T,T), (T,L)}

This experiment has 2 x 3 = 6 outcomes


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COUNTING TECHNIQUES


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Combination

counting the number of experimental

outcomes when n objects are to be selected

from a set ofN objects

where,

N! = N(N-1)(N-2) . . . (2)(1)

n! = n(n-1)(n-2) . . . (2)(1)

and, by definition, 0! = 1

!is called factorial


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Combination: EXAMPLE 1

How many ways can an HR manager randomly

hire two applicants from a set of five

applicants?


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Combination: EXAMPLE 2

A salesperson is going mall-to-mall to promote

her new juice drink. But for today, she can go

to only 3 of the 14 Pasig malls. How many sets

of 3 Pasig malls can she go to today?


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Permutation

counting the number of experimental

outcomes when n objects are to be selected

from a set ofN objects

but with regard to order

where,

N! = N(N-1)(N-2) . . . (2)(1)

n! = n(n-1)(n-2) . . . (2)(1)

and, by definition, 0! = 1


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Permutation: EXAMPLE 1

How many ways can an HR manager randomly

hire a product development specialist and a

advertising specialist from a set of five

applicants?


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Permutation: EXAMPLE 2

A salesperson is going mall-to-mall to promote

her new juice drink. But for this morning,

afternoon and evening, she can go to only

three of the 14 Pasig malls. How many wayscan she schedule her mall visits today?


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OTHER EXAMPLES


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1

You are studying the roof maintenance patterns

of the household heads at a barrio. You have

decided to stratify by appearance of the

household. In your very good appearance

stratum, there are 8 households / household

heads (HHs). From that particular group. . .


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1

QUESTIONS:

1. How many ways can you randomly select 2HHs?

2. How many ways can you randomly select 3

HHs?

3. How many ways can you randomly interview

3 HHs?


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2

How many ways can you arrange the letters of

the following words?

WATER

INTEGRAL

CURIOUS

BOOKKEEPER


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3

The CEO of a company is planning to have an

official executive photo for him, his two

female vice presidents and his two male vice

presidents. They are to line up for picturetaking.


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3

QUESTIONS:

1.How many arrangements are possible?2.How many arrangements are possible if...

a. One male VP insists on standing on the rightmost

side

b. One female VP insists on standing right next to

the CEO


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ASSIGNING PROBABILITIES


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Two basic requirements for

assigning probabilities

1. The probability assigned to each

experimental outcome must be between 0

and 1, inclusively. If we let Ei denote the ith

experimental outcome and P(Ei) its probability,then this requirement can be written as

(Anderson, Sweeney & Williams, 2008).


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Two basic requirements for

assigning probabilities

2. The sum of the probabilities for all the

experimental outcomes must equal 1. For n

experimental outcomes, this requirement can

be written as

(Anderson, Sweeney & Williams, 2008).


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Three methods of assigning

probabilities

1. Classical method all experimental outcomes are equallylikely

if there are n possible experimental outcomes, a probability of

1/n is assigned to each experimental outcome

EXAMPLES:

Tossing a fair coin

P(Head) = P(Tail) =

Rolling a fair die

P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6


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probabilities2. Relative frequency (or empirical)method appropriate when estimates on the

proportion of the time the experimental outcome will

occur if the experiment is repeated a large number of

times are available

EXAMPLE (Waiting time of patients as observed over a 20-

day period)

# of patients waiting # of days this outcome

occurred

0 2

1 5

2 6

3 44 3


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Relative frequency approximation:

Number of times an event occurred

___________________________

Number of times experiment was conducted


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probabilities

3. Subjective method

-- when little data are available

-- based on experience, intuition,

rumors, and/or belief

-- must still satisfy the two basic

requirements for assigningprobabilities


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EXAMPLES*

1. Suppose an experiment has five equally likely outcomes: E1, E2,

E3, E4, E5. Assign probabilities to each outcome. Which method

did you use?

2. A decision maker subjectively assigned the following

probabilities to four possible outcomes of an experiment:

P(E1) = .10, P(E2) = .15, P(E3) = .40, and P(E4) = .20

Are these probabilities valid?

*from Anderson, Sweeney, & Williams(2008)


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EXAMPLES*

3. An experiment with three outcomes has been repeated 50

times, and it was learned that E1 occurred 20 times, E2

occurred 13 times, and E3 occurred 17 times. Assign

probabilities to the outcomes. Which method did you use?

*from Anderson, Sweeney, & Williams(2008)


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EVENTS AND THEIR PROBABILITIES


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Event

Any subset of a sample space

A collection of sample points

An event is said to have occurred if any one of

its sample points appears as the experimental

outcome


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Probability of an Event

The sum of the probabilities of all the sample

points in the event

IfA is an event, the probability of eventA is

denoted by P(A)


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Example50 students in a class each specified the number of their siblings.

Following are the results:

Let Event A = the event of randomly picking a student with at most 3

siblings

Event B = . . . . . at least 5 siblings

What is P(A)? What is P(B)?

Number of siblings Number of students

who specified the

number on the left

1 3

2 9

3 13

4 13

5 8

6 4


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Any time that we can identify all the sample points of an

experiment and assign probabilities to each, we can compute

the probability of an event using the definition...

...However, in many experiments, the large number of sample

points makes the identification of the sample points, as well

as the determination of their associated probabilities,

extremely cumbersome, if not impossible

(Anderson, Sweeney, & Williams, 2008).


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Main references

Anderson, D.R., Sweeney, D.J., and Williams, T.A. (2008). Modern Business

Statistics with Microsoft Excel. Cincinnati, OH: South-Western/Thomson

Learning.

UP School of Statistics. (n.d.). Statistics 101 Handout / Course Notes. Diliman,

Quezon City.

Walpole, R. (1982). Introduction to Statistics. Singapore: Pearson Education.


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Basic Probability Concepts

(Part 1)

Basic Concepts in Probability

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