Probability

© 2012 Project Lead The Way, Inc.Principles Of Engineering

ProbabilityThe calculated likelihood that a given event will occur

Methods of Determining Probability

Empirical • An estimate that the event will happen

based on how often the event occurs from data collection or an experiment (large # of trials).

• Closely related to relative frequency. • Experimental observation

Example – Process control

Methods of Determining Probability

Theoretical• Number of ways an event can occur

divided by the total number of outcomes.• Uses known elements

Example – Coin toss, die rolling Subjective

AssumptionsExample – I think that . . .

In practice, engineers will often blend these approaches. For instance, engineers will assume that each one of the widgets produced at a factory has the same (unknown) chance of failure, then make observations to determine the likelihood of failure.

Methods of Determining Probability

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

Probability Components• For example, You might test a brass

sample to find its tensile strength. [experiment]

• It might break under any load from 0 to 1000 pounds in the tester, or it may not break at all. [sample space]

• One possible outcome is that a sample could break at 200 pounds. [Event]

• When you perform the test, it breaks at a particular load. [Outcome]

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

The probability of heads occurring on subsequent coin flips does not change.Notice that a binomial does not have to be a 50-50 chance. Getting a “6” on a die roll is also binomial (you either get the 6 or you don’t).

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

Bernoulli Process

• Technically a Bernoulli process happens only once (flip one coin), while a binomial process comes by adding many Bernoulli processes. The formula here is for a binomial process (combining the results of n independent Bernoulli trials).

• n! (read “n factorial”) is the product of all integers from 1 to n. For instance, 5! = 5 x 4 x 3 x 2 x 1 = 120

Bernoulli Process

• nCx (read “n choose x”) is the number of distinct groups of x things that can be chosen from n distinct things.

• Most scientific and graphing calculators have a ! key and a nCx key. Both functions can often be found in a probability menu.

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 !

(3-2)

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

*Independence means that one event’s outcome doesn’t affect the other event’s outcome.

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

*If A and B are mutually exclusive, it is not possible for both to occur at the same time. For example, you might want to know the probability of drawing either one of the two cards: a 2 of diamonds or a 2 of spades from a single draw.

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 the next card is a 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?