Download 4.1.1.A Probability

Probability
Principles Of Engineering
© 2012 Project Lead The Way, Inc.
Probability
The calculated likelihood that a given event
will occur
Methods of Determining Probability
Empirical
Experimental observation
Example – Process control
Theoretical
Uses known elements
Example – Coin toss, die rolling
Subjective
Assumptions
Example – I think that . . .
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
The result of an experiment
Probability
What is the probability of a tossed coin
landing heads up?
Experiment
Sample Space
Event
Outcome
Probability Tree
Probability
A way of communicating the belief that an
event will occur.
Expressed as a number between 0 and 1
fraction, percent, decimal, odds
Total probability of all possible events totals 1
Relative Frequency
The number of times an event will occur
divided by the number of opportunities
nx
fx =
n
fx
= Relative frequency of outcome x
n x = Number of events with outcome x
n = Total number of events
Expressed as a number between 0 and 1
fraction, percent, decimal, odds
Total frequency of all possible events totals 1
Probability
What is the probability of a tossed coin
landing heads up?
How many desirable
outcomes? 1
How many possible
outcomes? 2
fx
Px =
fa
1
P=
2
Probability Tree
=.5=50%
What is the probability of the coin
landing tails up?
Probability
What is the probability of tossing a coin
twice and it landing heads up both times?
HH
How many desirable
outcomes? 1
HT
How many possible
outcomes? 4
TH
fx
Px =
fa
1
P=
=.25=25%
4
TT
Probability
3rd
What is the probability of tossing
a coin three times and it landing
1
heads up exactly two times?
How many desirable
outcomes? 3
st
How many possible
outcomes? 8
2nd
HHH
HHT
HTH
HTT
THH
THT
fx
Px =
fa
3
P=
=.375=37.5%
8
TTH
TTT
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
Bernoulli Process
Px =
=
P=
(
)
(
)
( )
n!
x! n-x !
C
n x
p x qn-x
p x q n-x
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 Distribution
What is the probability of tossing a coin three
times and it landing heads up two times?
Px =
n!(px )(qn-x )
x!(n-x)!
3 × 2 ×1) × (0.5 ) (0.5 )
(
P=
(2 ×1) (1×1)
2
1
2
P2 = .375 = 37.50%
Law of Large Numbers
The more trials that are conducted, the closer
the results become to the theoretical probability
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%
Probability
AND (Multiplication)
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
Probability
AND (Multiplication)
What is the probability of rolling a 4 on a single die?
How many desirable outcomes? 1
1
P4 =
6
How many possible outcomes? 6
What is the probability of rolling a 1 on a single die?
How many desirable outcomes? 1
1
P1=
6
How many possible outcomes? 6
What is the probability of rolling a 4 and then a
1 in sequential rolls?
P=(P4 )(P1)
1 1
= ×
6 6
1
= =.0278=2.78%
36
Probability
OR (Addition)
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
Probability
OR (Addition)
What is the probability of rolling a 4 on a single die?
How many desirable outcomes? 1
1
P4 =
6
How many possible outcomes? 6
What is the probability of rolling a 1 on a single die?
How many desirable outcomes? 1
1
P1=
6
How many possible outcomes? 6
What is the probability of rolling a 4 or a 1 on a
single die?
1
1 2
P = P4 + P1 = + = = .3333 = 33.33%
6
6 6
Probability
NOT
Independent event not occurring
1 minus the probability of occurrence
P = 1 - P(A)
What is the probability of not rolling a 1 on a die?
1
5
P = 1 - P1 = 1 =
= .8333 = 83.33%
6
6
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
How many aces are in a deck? 4
How many face cards are in deck? 12
How many tens are in a deck? 4
Probability
What is the probability that the first card is an ace?
4
1
=
= .0769 = 7.69%
52
13
Since the first card was NOT a face, what is the
probability that the second card is a face card?
12
4
=
= .2353 = 23.53%
51 17
Since the first card was NOT a ten, what is the
probability that the second card is a ten?
4
= .0784 = 7.84%
51
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?
P=PA (PF +P10 )
1æ4 4ö
= ç + ÷
13 è 17 51ø
1 æ 12 4 ö
= ç + ÷
13 è 51 51ø
1 æ 16 ö
= ç ÷
13 è 51ø
= .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 Probability
P(E|A) = Probability of event E, given A
Example: One card is drawn from a shuffled
deck. The probability it is a queen is
4
1
P(queen) =
=
52 13
However, if I already know it is face card
P(queen | face)= 4 = 1
12 3
Conditional Probability
Probability 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’ Theorem
Calculates 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
=
P(E | A)× P(A)
P(E | A)× P(A) + P(E | B)× P(B) + P(E | C)× P(C)
Bayes’ Theorem 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
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
Known probabilities:
P(A)= 60%=.60
Probability the screen is from A
P(B)= 30%=.30
Probability the screen is from B
P(C)= 10%=.10
Probability the screen is from C
Bayes’ Theorem Example
Known conditional probabilities:
P(D|A)= 0.7%=.007
Probability the screen is defective
given it is from A
P(D|B)= 1.4%=.014
Probability the screen is defective
given it is from B
P(D|C)= 1.9%=.019
Probability the screen is defective
given it is from C
Bayes’ Theorem Example:
Defective Part
( )
P AD =
( ) ( )
P ( A ) ×P (D A ) +P (B) ×P (D B) +P (C) ×P (D C)
P A ×P D A
=
P(screen is defective AND from A)
P(screen is defective from anywhere)
LCD Screen Example
(.60)(.007)
P (A D) =
(.60)(.007)+(.30)(.014)+(.10)(.019)
.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?