Probability.pptx

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Probability

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Methods of Determining Probability

Empirical

Experimental observation

Example – Process control

Theoretical

Uses known elements

Example – Coin toss, die rolling

Subjective

Assumptions

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Probability

Components

Experiment

An activity with observable results

Sample Space

A set of all possible outcomes

Event

A subset of a sample space

Outcome / Sample Point

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Probability

What is the probability of a tossed coin

landing heads up?

Probability Tree

Experiment

Sample Space

Event

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Probability

A way of communicating the belief that an event will occur.

Expressed as a number between 0 and 1 fraction, percent, decimal, odds

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Relative Frequency

The number of times an event will occur divided by the number of opportunities

= Relative frequency of outcome x = Number of events with outcome x = Total number of events

Expressed as a number between 0 and 1 fraction, percent, decimal, odds

Total frequency of all possible events totals 1

x x

n

f

N



nfxx

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Probability

What is the probability of a tossed coin landing heads up?

How many possible outcomes? 2

How many desirable outcomes? 1

Probability Tree

What is the probability of the coin landing tails up?

x x

a

F P

F



1

2

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Probability

How many possible outcomes?

What is the probability of tossing a coin twice and it landing heads up both times?

4 HH HT TH TT x x a F P F 

1

4

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Probability

What is the probability of tossing a coin three times and it landing heads up exactly two times?

8 1st 2nd 3rd HHH HHT HTH HTT THH THT TTH TTT x x a

F

P

F



3

8

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Binomial Process

Each trial has only two possible outcomes

yes-no, on-off, right-wrong

Trial outcomes are independent

Tossing a coin does not affect future tosses

   





!

x n x

x

n p

q

P

x n x





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Bernoulli Process

P = Probability

x = Number of times an outcome occurs within n trials

n = Number of trials

p = Probability of success on a single trial q = Probability of failure on a single trial

   





!

x n x

x

n p

q

P

x n x





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Probability Distribution

What is the probability of tossing a coin three times and it landing heads up two times?

   



 



2 1

3×2×1× 0.5 0.5 P =

2×1 1×1

   





x n-x

x

n! p q P =

x! n - x !

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Law of Large Numbers

Trial 1: Toss a single coin 5 times H,T,H,H,T

P = .600 = 60%

Trial 2: Toss a single coin 500 times H,H,H,T,T,H,T,T,……T

P = .502 = 50.2%

Theoretical Probability = .5 = 50%

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Probability

Independent events occurring simultaneously Product of individual probabilities

If events A and B are independent, then the probability of A and B occurring is:

P = P(A)∙P(B)

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Probability

AND (Multiplication)

What is the probability of rolling a 4 on a single die?

6

What is the probability of rolling a 4 and then a 1 using two dice?

4 1 6 P  1 1 6 P  4 1

P = (P )(P ) = 1 1

6 6

1

.0278

36 2.78%

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Probability

Independent events occurring individually Sum of individual probabilities

If events A and B are mutually exclusive, then the probability of A or B occurring is:

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Probability

OR (Addition)

6

What is the probability of rolling a 4 or a 1 on a single die? 4 1 6 P  1 1 6 P  4 1 ( ) ( )

P  P  P 1 1

6 6

  2 .3333 33 3 %

6 . 3

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Probability

Independent event not occurring

1 minus the probability of occurrence

P = 1 - P(A)

NOT

What is the probability of not rolling a 1 on a die?

1

P   P 1 1

6

  5 .8333 83 3 %

6 . 3

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How many tens are in a deck?

Probability

Two cards are dealt from a shuffled deck.

What is the probability that the first card is an ace and the second card is a face card or a ten?

How many cards are in a deck? 52 4

12 4

How many aces are in a deck?

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Probability

What is the probability that the first card is an ace?

Since the first card was NOT a face, what is the probability that the second card is a face card?

Since the first card was NOT a ten, what is the probability that the second card is a ten?

4 1

.0769 7.69%

52 13  

12 4

.2353 23.53% 51 17  

4

.0784 7.84%

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Probability

Two cards are dealt from a shuffled deck.

What is the probability that the first card is an ace and the second card is a face card or a ten?

If the first card is an ace, what is the probability that the second card is a face card or a ten? 31.37%

1 4 4

= +

13 17 51

 

 

 

A F 10

P = P (P + P )

1 12 4

= +

13 51 51

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Bayes’ Theorem

The probability of an event occurring based upon other event probabilities

Notation Used:

P(A|E) = “Probability of A given E”

Probability of event A occurring given that we already know about E

 





 

1



1



 

2 I



2



I

 

n



n



P A • P E A

P A • P E A + P A • P E A + +P A • P E A



I



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LCD Screen Example

LCD screen components for a large cell phone manufacturing company are outsourced to three different vendors. Vendor A, B, and C supply 60%,

30%, and 10% of the required LCD screen components. Quality control experts have

determined that .7% of vendor A, 1.4% of vendor B, and 1.9% of vendor C components are defective.

If a cell phone was chosen at random and the LCD screen was determined to be

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LCD Screen Example

Notation Used:

P = Probability

D = Defective

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LCD Screen Example

1.List knowns:

Probability the screen is from A

Probability the screen is from B

Probability the screen is from C

P( )

A



P( )

B



P( )

C



60% .60



30% .30



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LCD Screen Example

Probability the screen is from A given that it is defective

Probability the screen is defective given it is from C

Probability the screen is defective given it is from A

Probability the screen is defective given it is from B

1.List knowns and unknowns:

P( | )

A D



P( | )

D C



P( | )

D A



P( | )

D B



0.7% .007



1.4% .014



1.9% .019



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LCD Screen Example





 





 





 

P A D =

P A P D A

P A P D A + P B P D B + P C P D C



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LCD

Screen

Example

  



  

   



.60 .007

.60 .007 + .30 .014 + .10 .019







P A D

.0042

.0042 .0042 .0019





.0042

.0103



.4078 40.78%

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LCD Screen Example

defective, what is the probability that the LCD screen was produced by vendor B?