Post on 03-Jan-2016
description
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?
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
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 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 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?
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 ) ( 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?