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

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

Px = Probability of outcome x

Fx = Frequency of outcome x

Fa = Absolute frequency of all events

xx

a

FP

F

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

Total probability of all possible events totals 1

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?


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



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

x n x

x

n! p qP

x! n x !

Bernoulli Process

x n x

x

n! p qP

x! n x !

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

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

2 13×2×1× 0.5 0.5P =

2×1 1×1

x n-x

x

n! p qP =

x! n - x !

P = .375 = 37.50%

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 = P(A) x P(B)

AND (Multiplication)

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


How many desirable outcomes? 16 4

1P

6

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



1P

6

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

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

= •6 6

1.0278

362.78%

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 = P(A) + P(B)

OR (Addition)

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



1P

6

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



1P

6

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

4 1P ( P ) ( P ) 1 16 6

2

.3333 33 3 %6

. 3

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 P 1

16

5

.8333 83 3 %6

. 3

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?

1 4 4= • +13 17 51

A F 10P = P (P + P )1 12 4

= • +13 51 51

1 16= •13 51

.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%

Bayes’ Theorem

I I

1 1 2 2 n n

P A • P E A

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

The probability of an event occurring based upon other event probabilities

IP A E =

LCD Screen 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?

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

P = Probability

D = Defective

A, B, and C denote vendors

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?

Probability

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