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Review Homework• Chapter 6: 1, 2, 3, 4, 13• Chapter 7 - 2, 5, 11
Probability Control charts for
attributes
Week 13 Assignment• Read Chapter 10:
“Reliability”• Homework
• Chapter 8: 5, 9,10, 20, 26, 33, 34
• Chapter 9: 9, 23
Week 12AgendaAgenda
Probability
Probability theoremsProbability theorems Probability is expressed as a number
between 0 and 1 Sum of the probabilities of the events of a
situation equals 1 If P(A) is the probability that an event will
occur, then the probability the event will not occur is• 1.0 - P(A)
Probability
Probability theoremsProbability theorems For mutually exclusive events, the
probability that either event A or event B will occur is the the sum of their respective probabilities.
When events A and B are not mutually exclusive events, the probability that either event A or event B will occur is • P(A or B or both) = P(A) + P(B) - P(both)
Probability
Probability theoremsProbability theorems If A and B are dependent events, the
probability that both A and B will occur is • P(A and B) = P(A) x P(B|A)
If A and B are independent events, then the probability that both A and B will occur is• P(A and B) = P(A) x P(B)
Probability
Permutations and Permutations and combinationscombinations
A permutation is the number of arrangements that n objects can have when r of them are used.
When the order in which the items are used is not important, the number of possibilities can be calculated by using the formula for a combination.
Probability
Discrete probability Discrete probability distributionsdistributions
Hypergeometric - random samples from small lot sizes.• Population must be finite• samples must be taken randomly without
replacement Binomial - categorizes “success” and
“failure” trials Poisson - quantifies the count of discrete
events.
Probability
Continuous probability Continuous probability distributionsdistributions
Normal Uniform Exponential Chi Square F student t
Probability
Fundamental conceptsFundamental concepts Probability = occurrences/trials 0 < P < 1 The sum of the simple probabilities for all
possible outcomes must equal 1 Complementary rule - P(A) + P(A’) = 1
Probability
Addition ruleAddition rule P(A + B) = P(A) + P(B) - P(A and B)
• If mutually exclusive; just P(A) + P(B)
P(A) P(B)
P(AandB)
Probability
Addition rule exampleAddition rule example P(A + B) = P(A) + P(B) - P(A and B) Roll one die
• Probability of even and divisible by 1.5?• Sample space {1,2,3,4,5,6}
• Event A - Even {2,4,6}• Event B - Divisible by 1.5 {3,6}• Event A and B ?
Solution?
Probability
Conditional probability ruleConditional probability rule P(A|B) = P(A and B) / P(B) A die is thrown and the result is known to
be an even number. What is the probability that this number is divisible by 1.5?• P(/1.5|Even)=P(/1.5 and even)/P(even)• 1/6 / 3/6 = 1/3
Probability
Compound or joint Compound or joint probabilityprobability
The probability of the simultaneous occurrence of two or more events is called the compound probability or, synonymously, the joint probability.
Mutually exclusive events cannot be independent unless one of them is zero.
Probability
Multiplication for Multiplication for independent eventsindependent events
P(A and B) = P(A) x P(B)
• P(ace and heart) = P(ace) x P(heart)
• 1/13 x 1/4 = 1/52
Probability
Computing conditional Computing conditional probabilitiesprobabilities
P(A|B) = P(A and B)/P(B) P(Own and Less than 2 years)?
Number of credit applicants by category
On present job 2years or less
On present jobmore than 2 years
Own Home 20 40
Rent Home 80 60
Total 100 100
Probability
P(A) P(B)
P(AandB)
Computing conditional Computing conditional probabilitiesprobabilities
P(A|B) = P(A and B)/P(B)
Probability
J oint probabilitytable
On present job2 years or less
On present jobmore than 2
years
Marginalprobability
Own Home .10 .20 .30
Rent Home .40 .30 .70
Marginal probability .50 .50 1.00
Probability
Conditional probabilityConditional probability
Satisfied Not Satisfied Totals
New 300 100 400 Used 450 150 600
Total 750 250 1000
S=satisfied N= bought new car
P(N|S) = ?
Probability
Just for funJust for fun 60 business students
from a large university are surveyed with the following results:• 19 read Business Week• 18 read WSJ• 50 read Fortune• 13 read BW and WSJ• 11 read WSJ and Fortune• 13 read BW and Fortune• 9 read all three
How many read none? How many read only
Fortune? How many read BW, the
WSJ, but not Fortune?
Hint: Try a Venn diagram.
Probability
Learning objectivesLearning objectives Know the difference between discrete
and continuous random variables. Provide examples of discrete and
continuous probability distributions. Calculate expected values and
variances. Use the normal distribution table.
Probability
Random variablesRandom variables A random variable is a numerical quantity
whose value is determined by chance.• “A random variable assigns a number to
every possible outcome or event in an experiment”.
• For non-numerical outcomes such as a coin flip you must assign a random variable that associates each outcome with a unique real number.
Probability
Random variable typesRandom variable types Discrete random variable - assumes a
limited set of values; non-continuous, generally countable• number of Mark McGwire homeruns in a
season• number of auto parts passing assembly-line
inspection• GRE exam scores
Probability
Random variable typesRandom variable types Continuous random variable - random
variable with an infinite set of values.
Can occur anywhere on a continuous number scale
0.000 1.000Baseball player’s batting average
Probability
Random variables and Random variables and probability distributionsprobability distributions
The relationship between a random variable’s values and their probabilities is summarized by its probability distribution.
Probability
Probability distributionProbability distribution Whether continuous or discrete, the
probability distribution provides a probability for each possible value of a random variable, and follows these rules:• The events are mutually exclusive• The individual probability values are
between 0 and 1.• The total value of the probability values sum
to 1
Probability
Probability distribution for Probability distribution for rates of returnrates of return
Possible rate of return• 10%• 11%• 12%• 13%• 14%• 15%• 16%• 17%
Probability• .05• .15• .20• .35• .10• .10• .03• .02
Total = 1.0
Probability
Describing distributionsDescribing distributions Measures of
central tendency• expected value
• (weighted average)
Measures of variability• variance• standard deviation
Probability
Expected value of a discrete Expected value of a discrete random variablerandom variable
For discrete random variables, the expected value is the sum of all the possible outcomes times the probability that they occur.
E(X) = {xi * P(xi)}
Probability
Example: A fair dieExample: A fair die Roll 1 die: x P(x) x*P(x) E(x)=?
1 1/6 1/6
2 1/6 2/6
3 1/6 3/6
4 1/6 4/6
5 1/6 5/6
6 1/6 6/6
Can you sketch the distribution?
Probability
Fair die illustrates a discrete Fair die illustrates a discrete “uniform distribution”“uniform distribution”
The random variable, x, has n possible outcomes and each outcome is equally likely. Thus, x is distributed uniform.
Probability
Example: An unfair dieExample: An unfair die Roll 1 die: x P(x) x*P(x) E(x)=?
1 1/12 1/12
2 2/12 4/12
3 2/12 6/12
4 2/12 8/12
5 2/12 10/12
6 3/12 18/12
Can you sketch the distribution?
Probability
Expected value of a betExpected value of a bet Suppose I offer you the following wager:
You roll 1 die. If the result is even, I pay you $2.00. Otherwise you pay me $1.00.
E(your winnings)=.5 ($2.00) + .5 (-1.00)
= 1.00 - .50 = $0.50
Probability
Expected Value of a BetExpected Value of a Bet Suppose I offer you the following wager:
You roll 1 die. If the result is 5 or 6 I pay you $3.00. Otherwise you pay me $2.00.
What is your expected value?
Probability
Variance of a discrete Variance of a discrete random variablerandom variableThe variance of a random variable is a
measure of dispersion calculated by squaring the differences between the expected value and each random variable and multiplying by its associated probability.
{(xi-E(x))2 * P(xi)}
Probability
Roll 1 die: [x- E(X)] 2 P(x) *P(x)
1 - 21/6 6.25 1/6 1.04
2 - 21/6 2.25 1/6 .375
3 - 21/6 .25 1/6 .04
4 - 21/6 .25 1/6 .04
5 - 21/6 2.25 1/6 .375
6 - 21/6 6.25 1/6 1.042.91
Example: A fair dieExample: A fair die
Probability
Probability distributions for Probability distributions for continuous random variablescontinuous random variables
A continuous mathematical function describes the probability distribution.
It’s called the probability density function and designated ƒ(x)
Some well know continuous probability density functions:• Normal Beta• Exponential Student t
Probability
Continuous probability Continuous probability density function - Uniform density function - Uniform If a random variable, x, is distributed
uniform over the interval [a,b], then its pdf is given by
f xb a
( ) 1
a b
1 b-a
Probability
UniformUniform
a b
1 b-a
c
P(c<x<b) = Area of brown rectangle 1 * (b-c) Ht x Width)= b-a
Probability
Uniform distribution Uniform distribution
If a random variable, x, is distributed uniform over the interval [a,b], then its pdf is given by
f xb a
( ) 1
And, the mean and variance are (a+b) ( b-a )2
E(x) = ------- Var(x)=--------- 2 12
Probability
f x( ) 1
5
And, the mean and variance are (a+b) ( b-a )2 25E(x) = ------ = 5.5 V(x)=--------- = ----- = 2.08 2 12 12
So, if a = 3 and b = 8
Calculate uniform mean, Calculate uniform mean, variancevariance
Probability
Continuous pdf - NormalContinuous pdf - Normal
f x ex
( )( )
1
2 2
2
2
2
If x is a normally distributed variable, then
is the pdf for x. The expected value is and the variance is 2.