1. Operations Management
Session 10: Probability Concepts

2. Operations Management
Simulation Game
Game codes due.
Please go to to register.
Course code: usc.
Individual code: what you purchased from the bookstore.
Case groups posted. Please double-check.
Session 10
Operations Management

3. Operations Management
Today’s Class
Probability Concept Review
Basic Statistics Formula
Common Distribution
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Operations Management

4. Operations Management
Quote of the day
Without the element of uncertainty, the bringing off of even, the greatest business triumph would be dull, routine, and eminently unsatisfying.
J. Paul Getty
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5. Operations Management
Blackjack
You have a 9 and 5, what will happen if you hit?
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6. Operations Management
Random Experiment
Random Experiment: An experiment in which the precise outcome is not known ahead of time. The set of possibilities however is known
Examples:
Demand for blue blazers next month
The value of a rolled die
The waiting times of customers in the bank
The waiting time for an ATT service person
Tomorrow’s closing value of the NASDAQ
The temperature in Los Angeles tomorrow
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Operations Management

7. Operations Management
Random Variable
A random variable is the numerical value determined by the outcome of a random experiment
A random variable can be discrete (i.e. takes on only a finite set of values) or continuous
Examples:
The value on a rolled die is a discrete random variable
The demand for blazers is a discrete random variable
The birth weight of a newborn baby is a continuous variable
The waiting time for the AT&T service person is a continuous random variable
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8. Operations Management
Sample Space
Sample space is the list of possible outcomes of an experiment
Examples:
For a die, the sample space S is: {1,2,3,4,5,6}
For the demand for blue blazers it is all possible realizations of the demand. For example: {1000,1001,1002…,2000}
The waiting time in the bank is any number greater than or equal to 0. This is a continuous random variable
The waiting time for a bus at a bus stop is any number between 0 and 30 minutes. This is a continuous random variable that is bounded
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9. Operations Management
Event
An event is a set of one or more outcomes of a random experiment
Examples:
Getting less than 5 by rolling the die: This event occurs if the values observed are {1,2,3, or 4}
The demand is smaller or equal to This event occurs if the values of the demand are {1000, 1001, … 1500}
The waiting time for a bus at the bus stop exceeds 10. This event occurs if the wait time is in the interval (10, 30)
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10. Operations Management
Probability
The probability of an event is a number between 0 and 1
1 means that the event will always happen
0 means that the event will never happen
The probability of an event A is denoted as either P(A) or Prob(A)
Example: Probability of Rolling die and observing a number less than 5 =
P(outcome< 5) = Prob(observing {1,2,3 or 4}) = 4/6 = 2/3
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11. Operations Management
Probability
Probability that A doesn’t occur:
P(not A) = 1 – P(A)
Thus, the probability you will roll a number larger or equal to 5 is or 6 is: 1 – Probability (Outcome <5) = 1 – 2/3 = 1/3
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12. Operations Management
Probability
Suppose all the outcomes that constitute the “waiting time” for an AT&T operator are equally likely. The minimum waiting time is 30 min and the maximum is 90 min.
Then the probability of waiting less than 45 min is:
P (waiting more than 45 min) is:
Event
Sample Space
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13. Probability Distribution for Discrete Random Variables
Let us begin with discrete outcomes
A probability distribution is a list of:
All possible values for a random variable (Sample space); and
The corresponding probabilities
For a die,
the probability distribution is:
Outcome
Probability 1
1/6 2 3 4 5 6
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14. Probability Distribution for Discrete Random Variables
The chart below depicts the probability distribution
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15. Cumulative Probability Distribution for Discrete Random Variables
Probability that a random number will be less than or equal to some given number
For a die, the cumulative
probability distribution is:
Outcome less than or equal to:
Probability 1
1/6 2
2/6 3
3/6 4
4/6 5
5/6 6
Additional: What is the
probability a die roll is
less than 3.5?
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Operations Management

16. Continuous Random Variables and Probability Density Functions (PDF)
The probability density function is the analog of the probability distribution (table 1) for discrete random numbers
Example: Suppose we have a computer program that can generate any number between 1 and 6 (not just the integers)
Assume that each number is equally likely to be generated. Then we have a continuous random number
This random number has a uniform distribution between 1 and 6
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17. Continuous Random Variables and Probability Density Functions (PDF)
1/5 1 2 3 4 5 6
Outcome
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18. Properties of Probability Density Functions
By convention the total area under the probability density function must equal 1
The base of the rectangle in the figure is 6 – 1 = 5 units long, the probability density is 1/5 for all values between 1 and 6. This ensures that the total area is 1
The probability of observing any value between two numbers is equal to the area under the probability density function between those numbers
The probability of observing any number between 4.0 and 5.0 will be (5.0 – 4.0)* 1/5 = 1/5 = 0.2
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19. Properties of Probability Density Functions
1/5 1 2 3 4 5 6
Outcome
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20. Properties of Cumulative Distribution Functions
CDF, F
Probability that the
outcome is smaller
than 5: is 4/5 1
4/5
Probability that the
outcome is smaller
than 2: is 1/5
1/5 1 2 3 4 5 6
Outcome
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21. Relationship between Density and Cumulative Distribution Functions
CDF 1
4/5
1/5 1 2 3 4 5 6
Outcome
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22. Other distributions: Triangular
Probability that the outcome
is between 2 and 5 2 5
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23. Operations Management
Normal Distribution
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80 -4 -3 -2 -1 1 2 3 4 X
Normal distribution #1
Normal distribution #2
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24. Continuous Random Variables and Probability Density Functions (PDF)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80 -4 -3 -2 -1 1 2 3 4 X
Normal distribution #1
Normal distribution #2
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25. Cumulative Density Function (CDF) for Continuous Random Numbers
This is analogous to the cumulative distribution function for discrete random numbers
The cumulative density function gives the probability of the continuous random variable being equal to or smaller than a given number
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26. Cumulative Density Function (CDF) for Continuous Random Numbers1
1/5 o 1 2 3 4 5 6
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27. Mean or Expected Value of a Random Number
Expected value can be thought of as the average value of a random variable
Let us denote by X the value of the random variable.
If the random variable is the value of a die, then X denotes the value rolled. If we roll a 6, then X = 6). We will use the notation E[X] to denote the expected value of X
If the random number is a discrete variable that can take on values between 1 and N then:
E[X] = Thus for the die, E[X] = 1/6*1 + 1/6*2 + 1/6*3 + 1/6*4 + 1/6*5 + 1/6*6 = 3.5
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28. Mean or Expected Value of a Random Number
What if the variable is a continuous random variable?
Let f(X) be the probability density function.
Example: for the uniform distribution, we have seen:
f(X) = 0.2 whenever X is between 1 and 6. f(X) = 0 if X is not between 1 and 6.]
Integration of continuous variables in lay terms is equivalent to summation for discrete variables.
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29. Operations Management
The Variance of X
When X is a discrete random variable:
Var(X) =  (X – E[X])2*Prob(X)
If X is the random number generated by the roll of a die then:
Var(X) = (1-3.5)2*1/6 + (2-3.5)2*1/6 +(3-3.5)2*1/6 +(4-3.5)2*1/6 +(5-3.5)2*1/6 +(6-3.5)2*1/6 =
Standard Deviation = square root of variance
SD(X) = in this example
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30. How to measure variability?
A possible measure is variance, or standard deviation
Is this good enough?
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31. Which one has the larger variability?
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32. Which one has the larger variability?
The variation in the first set appears to be significantly higher than the second set.
Nevertheless, the standard deviation of the first graph is 5, the standard deviation of the second graph is 10.
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33. Coefficient of Variation
A better measure of variability is the ratio of the standard deviation to the average. This ratio is called the coefficient of variation.
Coefficient of Variation = Standard Deviation / Average (expected value)
A similar measure is squared coefficient of variation:
SCV = (CV)2 = (SD/M)2
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34. Operations Management
Sum of Random Numbers
Often we have to analyze sum of random numbers.
Examples include:
The sum of the demand of different products processed by the same resource
The total demand for cars produced by GM
The total demand for knitwear at DD
The total completion time of a project
The sum of throughput times at two different stages of a service system (waiting time to place an order at a cafeteria and waiting time in the line to pay for the food)
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35. Operations Management
Sum of Random Numbers
Let X and Y be two random variables. The sum of X and Y is another random variable. Let S = X +Y
The distribution of S will be different from that of X and Y
Example:
Let S be the sum of the values when you roll 2 dice simultaneously. Let X represent the value die #1 and Y represent the value of die #2
S = X + Y
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36. Operations Management
Sum of Random Numbers
The distribution of the sum S is given below: S
Prob(S) 2
1/36 7
6/36 3
2/36 8
5/36 4
3/36 9
4/36 5 10 6 11 12
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37. Operations Management
Sum of Random Numbers
E[S] = 2*1/36 + 3*2/36 + 4*3/36 + 5*4/36 + 6*5/36 + 7*6/36 + 8*5/36 + 9*4/ *3/ *2/ *1/36 = 7
Var(S) = (2 - 7)2*1/36 + (3 - 7)2*2/36 +…….+ (12 - 7)2*1/36 = 5.83
SD(S) = 5.83^1/2 = 2.42
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38. Operations Management
Sum of Random Numbers
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39. Expected Value and Standard Deviation of Sum of Random Numbers
If a and b are 2 known constant and X and Y are random independent variables:
E[aX+bY] = aE[X] + bE[Y]
Var(aX+bY) = a2Var(X) + b2Var(Y)
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40. Specific Distributions Of Interest
We will also utilize Uniform Distributions
Uniform Distribution: Whenever the likelihood of observing a set of numbers is equally likely
Continuous or discrete
We use notation U(a,b) to denote a uniform distribution
Example U(1,5) is uniform distribution between 1 and 5.
If it is a discrete distribution then outcomes 1,2,3,4, and 5 are equally likely (each with probability 1/5)
If it is a continuous distribution then all numbers between 1 and 5 are equally likely
The p.d.f. for U(1,5) (continuous) will be f(X) = 0.25 for X between 1 and 5
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41. Exponential Distribution
The exponential distribution is often used as a model for the distribution of time until the next arrival.
The probability density function for an Exponential distribution is: f(x) = e-x, x > 0
 is a parameter of the model (just as m and s are parameters of a Normal distribution)
E[X] = 1/ Var(X) = 1/2
Coefficient of Variation = Standard deviation / Average = 1
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42. Exponential Distribution
Shape of the Exponential Probability Density Function
f(X) X
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43. Operations Management
Poisson Distribution
The Poisson Distribution is often used as a model for the number of events (such as the number of telephone calls at a business or the number of accidents at an intersection) in a specific time period
The probability of n events is: p(n) = ne-/n!, n = 0, 1, 2, 3, …
 is a parameter of the model
E[N] = 
Var(N) = 
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44. Operations Management
Poisson Distribution
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45. Operations Management
Next Class
Waiting-line Management
How uncertainty/variability and utilization rate determines the system performance
Article Reading: “The Psychology of Waiting-lines”
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Operations Management