Probability

ProbabilityThe 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 . . .

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

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

Probability Tree

Experiment

Sample Space

Event

Outcome

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

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

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?

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%

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%

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

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

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 !

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

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)

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%

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)

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%

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%

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?

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

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%

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)=

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)

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

=

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?

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

?

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

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

Bayes’ Theorem Example:Defective Part

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

LCD Screen Example

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

.60 .007=

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

.0042=

.0042+.0042+.0019

.0042=

.0103

= .4078 = 40.78%

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?