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Page 1: Probability. The calculated likelihood that a given event will occur.

Probability

Page 2: Probability. The calculated likelihood that a given event will occur.

ProbabilityThe calculated likelihood that a given event will occur

Page 3: Probability. The calculated likelihood that a given event will occur.

Methods of Determining Probability

Empirical

Experimental observationExample – Process control

TheoreticalUses known elements

Example – Coin toss, die rolling Subjective

AssumptionsExample – I think that . . .

Page 4: Probability. The calculated likelihood that a given event will occur.

Probability Components

ExperimentAn activity with observable results

Sample SpaceA set of all possible outcomes

EventA subset of a sample space

Outcome / Sample PointThe result of an experiment

Page 5: Probability. The calculated likelihood that a given event will occur.

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

Probability Tree

Experiment

Sample Space

Event

Outcome

Page 6: Probability. The calculated likelihood that a given event will occur.

Probability

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

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

Total probability of all possible events totals 1

Page 7: Probability. The calculated likelihood that a given event will occur.

Relative FrequencyThe number of times an event will occur divided by the number of opportunities

= Relative frequency of outcome x

= Number of events with outcome x

n = Total number of events

xx

nf =

n

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

Total frequency of all possible events totals 1

xn

Page 8: Probability. The calculated likelihood that a given event will occur.

Probability

xx

a

fP =

f

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

How many possible outcomes? 2

How many desirable outcomes? 1

1P=

2=.5=50%

Probability Tree

What is the probability of the coin landing tails up?

Page 9: Probability. The calculated likelihood that a given event will occur.

Probability

xx

a

fP =

f

How many possible outcomes?

How many desirable outcomes? 1

1P=

4

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

4

HH

HT

TH

TT

=.25=25%

Page 10: Probability. The calculated likelihood that a given event will occur.

Probability

xx

a

fP =

f

How many possible outcomes?

How many desirable outcomes? 3

3P=

8

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

=.375=37.5%

Page 11: Probability. The calculated likelihood that a given event will occur.

Binomial Process

Each trial has only two possible outcomesyes-no, on-off, right-wrong

Trial outcomes are independent Tossing a coin does not affect future tosses

Page 12: Probability. The calculated likelihood that a given event will occur.

Bernoulli Process

P = Probability

x = Number of times for a specific outcome within n trials

n = Number of trials

p = Probability of success on a single trial

q = Probability of failure on a single trial

! = factorial – product of all integers less than or equal

Page 13: Probability. The calculated likelihood that a given event will occur.

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

( )( )( )

x n-x

x

n! p qP =

x! n-x !

Page 14: Probability. The calculated likelihood that a given event will occur.

Law of Large Numbers

Trial 1: Toss a single coin 5 times H,T,H,H,TP = .600 = 60%

Trial 2: Toss a single coin 500 times

H,H,H,T,T,H,T,T,……TP = .502 = 50.2%

Theoretical Probability = .5 = 50%

The more trials that are conducted, the closer the results become to the theoretical probability

Page 15: Probability. The calculated likelihood that a given event will occur.

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(A and B) = PA∙PB

AND (Multiplication)

Page 16: Probability. The calculated likelihood that a given event will occur.

Probability AND (Multiplication)What is the probability of rolling a 4 on a single die?

How many possible outcomes?

How many desirable outcomes? 16

4

1P =

6

What is the probability of rolling a 1 on a single die?

How many possible outcomes?

How many desirable outcomes? 16 1

1P =

6

What is the probability of rolling a 4 and then a 1 in sequential rolls?

4 1P=(P )(P )1 1

=6 6×

1= =.027 28=

36.78%

Page 17: Probability. The calculated likelihood that a given event will occur.

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:

P(A or B) = PA + PB

OR (Addition)

Page 18: Probability. The calculated likelihood that a given event will occur.

Probability OR (Addition)What is the probability of rolling a 4 on a single die?

How many possible outcomes?

How many desirable outcomes? 16

4

1P =

6

What is the probability of rolling a 1 on a single die?

How many possible outcomes?

How many desirable outcomes? 16 1

1P =

6

What is the probability of rolling a 4 or a 1 on a single die?

4 1P = P + P1 1

= + 6 6

2= = .333 333 =

6.33%

Page 19: Probability. The calculated likelihood that a given event will occur.

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?

1P = 1 - P1

= 1 - 6

5= = .833 833 =

6.33%

Page 20: Probability. The calculated likelihood that a given event will occur.

How many tens are in a deck?

ProbabilityTwo 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?

How many face cards are in deck?

Page 21: Probability. The calculated likelihood that a given event will occur.

Probability

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

4 1 = = .0769 = 7.69%

52 13

12 4 = = .2353 = 23.53%

51 17

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 = .0784 = 7.84%

51

Page 22: Probability. The calculated likelihood that a given event will occur.

ProbabilityTwo 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?

A F 10P=P (P +P )

= .0241 = 2.41%

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

Page 23: Probability. The calculated likelihood that a given event will occur.

Conditional ProbabilityP(E|A) = Probability of event E, given A

Example: One card is drawn from a shuffled deck. The probability it is a queen is

P(queen) =

However, if I already know it is face card

P(queen | face)=

Page 24: Probability. The calculated likelihood that a given event will occur.

Conditional ProbabilityProbability of two events A and B both occurring =

P(A and B)

= P(A|B) P(B)

= P(B|A) P(A)

If A and B are independent, then

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

Page 25: Probability. The calculated likelihood that a given event will occur.

Bayes’ TheoremCalculates a conditional probability, based on all the ways the condition might have occurred.

P( A | E ) = probability of A, given we already know the condition E

=

Page 26: Probability. The calculated likelihood that a given event will occur.

Bayes’ Theorem ExampleLCD 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 defective, what is the probability that the LCD screen was produced by vendor A?

Page 27: Probability. The calculated likelihood that a given event will occur.

Bayes’ Theorem Example

Notation Used:

P = Probability

D = Defective

A, B, and C denote vendors

Unknown to be calculated:

P(A|D)= Probability the screen is from A,given that it is defective

?

Page 28: Probability. The calculated likelihood that a given event will occur.

Bayes’ Theorem Example

P(A)=

P(B)=

P(C)=

Known probabilities:

Probability the screen is from A

Probability the screen is from B

Probability the screen is from C

60%=.60

30%=.30

10%=.10

Page 29: Probability. The calculated likelihood that a given event will occur.

Bayes’ Theorem Example

P(D|C)=Probability the screen is defective given it is from C

P(D|A)=

P(D|B)=

Probability the screen is defective given it is from A

Probability the screen is defective given it is from B

Known conditional probabilities:

0.7%=.007

1.4%=.014

1.9%=.019

Page 30: Probability. The calculated likelihood that a given event will occur.

Bayes’ Theorem Example:Defective Part

= P(screen is defective AND from A) P(screen is defective from anywhere)

Page 31: Probability. The calculated likelihood that a given event will occur.

LCD Screen Example

( )( )( )( ) ( )( ) ( )( )

.60 .007=

.60 .007 + .30 .014 + .10 .019( )P A D

.0042=

.0042+.0042+.0019

.0042=

.0103

= .4078 = 40.78%

Page 32: Probability. The calculated likelihood that a given event will occur.

LCD Screen Example

If a cell phone was chosen at random and the LCD screen was determined to be defective, what is the probability that the LCD screen was produced by vendor B?

If a cell phone was chosen at random and the LCD screen was determined to be defective, what is the probability that the LCD screen was produced by vendor C?