operations management
Class 14 Intro. to Probability and Excel
Instructor: Mani Lakshmanan
P300 Introduction to Operations Management
Random Event
An outcome with chance: we know what outcomes could happen, but we don’t know which particular outcome did or will happen.
Winner of the football game: IU v.s. Perdue
Possible Outcome = {IU, Perdue}
The side you get after tossing a coin
Possible Outcome = {Head, Tail}
The number you get after tossing a six-sided die
Possible Outcome = {1, 2, 3, 4, 5, 6}
Sales of iPhone 6plus tomorrow through Apple’s official website
Possible Outcome = {0, 1, 2, 3, … }
Possible outcomes are distinct numbers.
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Random Event
An outcome with chance: we know what outcomes could happen, but we don’t know which particular outcome did or will happen.
Lowest temperature () at Bloomington in 2017
Possible Outcome = Any value between -30 and 0,
i.e., [-30,0]
The amount of rainfall at a particular site next month
Possible Outcome = Any nonnegative value,
i.e.,
Possible outcomes take values continuously in an interval.
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Probability
Probability
How likely something is to happen
The side you get after tossing a coin
Possible Outcome = {Head, Tail}
The probability of the coin landing Head is ½.
And the probability of the coin landing Tail is ½.
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Chance
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Question:
What is the number after tossing a six-sided die?
Answer: ?
The number can be 1, 2, 3, 4, 5, or 6, the probability of taking each of the six numbers is 1/6.
The number < 1 with probability (w.p.) 0;
<=1, 1.1, 1.2, … 1.99 w.p. 1/6;
<=2, 2.1, 2.2, … 2.99 w.p. 2/6;
<=3 … w.p. 3/6;
<=4 … w.p. 4/6;
<=5 … w.p. 5/6;
<=6 … w.p. 1 .
How do we describe above Q & A by math?
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Discrete Random Variable,
Question:
What is the number after tossing a six-sided die?
Possible outcomes are distinct numbers: 1,2,3,…,6
We can use a discrete random variable to represent “the number after tossing a six-sided die”, i.e.,
= the number after tossing a six-sided die
Definition: A discrete random variable represents a random event whose possible outcomes are distinct numbers
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=the number after tossing a six-sided die
Question:
What is the number after tossing a six-sided die?
What is ?
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=the number after tossing a six-sided die
Question:
What is the number after tossing a six-sided die?
Answer: ?
The number can be 1, 2, 3, 4, 5, or 6, the probability of taking each of the six numbers is 1/6.
Probability mass function (p.m.f.): assign probability to each possible outcome
for ,6
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=the number after tossing a six-sided die
Question:
What is the number after tossing a six-sided die?
Answer:
The number can be 1, 2, 3, 4, 5, or 6, the probability of taking each of the six numbers is 1/6.
Answer:
follows for
Question: What is ?
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Question:
What is the number after tossing a six-sided die?
Answer: ?
The number can be 1, 2, 3, 4, 5, or 6, the probability of taking each of the six numbers is 1/6.
The number < 1 with probability (w.p.) 0;
<=1, 1.1, 1.2, … 1.99 w.p. 1/6;
<=2, 2.1, 2.2, … 2.99 w.p. 2/6;
<=3 … w.p. 3/6;
<=4 … w.p. 4/6;
<=5 … w.p. 5/6;
<=6 … w.p. 1 .
How do we describe above Q & A by math?
10
=the number after tossing a six-sided die
The number < 1 with probability (w.p.) 0;
<=1, 1.1, 1.2, … 1.99 w.p. 1/6;
<=2, 2.1, 2.2, … 2.99 w.p. 2/6;
<=3 … w.p. 3/6;
<=4 … w.p. 4/6;
<=5 … w.p. 5/6;
<=6 … w.p. 1 .
=
Cumulative distribution function (c.d.f. ): is the probability that is smaller than or equal to a certain number
1
1
2
3
4
5
6
1/6
2/6
0
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=the number after tossing a six-sided die
Question:
What is the number after tossing a six-sided die?
Answer: ?
The number can be 1, 2, 3, 4, 5, or 6, the probability of taking each of the six numbers is 1/6.
The number < 1 with probability (w.p.) 0;
<=1, 1.1, 1.2, … 1.99 w.p. 1/6;
<=2, 2.1, 2.2, … 2.99 w.p. 2/6;
<=3 … w.p. 3/6;
<=4 … w.p. 4/6;
<=5 … w.p. 5/6;
<=6 … w.p. 1 .
follows for
What is ?
follows …
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Discrete Random Variable,
The number of heads in two coin tosses
Probability mass function (p.m.f.): assign probability to each possible outcome
,
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Discrete Random Variable,
The number of heads in two coin tosses
Cumulative distribution function (c.d.f. ): is the probability that is smaller than or equal to a certain number
=
1
0
1
2
0.25
0.75
0
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Discrete Random Variable,
A discrete random variable represents a random event whose possible outcomes are distinct numbers
The number you get after tossing a six-sided dice
Possible Outcome = {1, 2, 3, 4, 5, 6}
The number of heads in two coin tosses
Possible Outcome = {0,1,2}
The sum of two dice outcomes.
Possible Outcome = {2, 3, 4, … , 12}
Sales of iPhone 5s tomorrow through Apple’s official website;
Possible Outcome = {0, 1, 2, 3, … }
The number of customers visit a given store in one hour
Possible Outcome = {0, 1, 2, 3, … }
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Continuous Random Variable,
Random event with possible outcomes that take values continuously in an interval
Lowest temperature (℉) at Bloomington in 2015
Possible Outcome = Any value between -30 and 0,
i.e., [-30,0]
The amount of rainfall at a particular site next month
Possible Outcome = Any nonnegative value,
i.e.,
The lifetime of a light bulb
Possible Outcome = Between 0 and 100 hours,
i.e., [0, 100]
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=The lifetime of a light bulb
Possible outcome = [0,100]
The bulb can last less than or equal to hours ( is a number between 0 and 100hrs) with probability
For example, the bulb can last at most 20 hours with probability .
Cumulative distribution function (c.d.f. ): is the probability that is smaller than or equal to a certain number
Probability density function (p.d.f.):
such that
Similar to p.m.f. for discrete random variable
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Exponential random variable,
Continuous random variable takes values in
Defined by one parameter:
p.d.f.
c.d.f.
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Normal random variable,
Continuous random variable takes values in
Defined by two parameters:
c
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Normal random variable,
follows a normal distribution
What is the probability that is less than or equal to 102?
Method 1:
Excel function: =NORM.DIST(102, 100,true)
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Normal random variable,
follows a normal distribution
What is the probability that is less than or equal to 102?
Method 2:
Transformation:
Read from the table of cumulative distribution of the standard normal r.v.
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Normal random variable,
follows a normal distribution
is smaller or equal to what number with probability 0.8?
Method 1:
Excel function: =NORM.INV(0.8, 100)
Excel return 112.6
Thus, is smaller than or equal to 112.6 with probability 0.8
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Normal random variable,
follows a normal distribution
is smaller or equal to what number with probability 0.8?
Method 2:
Step 1. Find z-value that is closest to 0.8
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Normal random variable,
follows a normal distribution
is smaller or equal to what number with probability 0.8?
Method 2:
Step 1. Find z-value that is closest to 0.8
Z=0.84
Step 2.
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Mean of a Random Variable
Refers to the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained.
More formally, the expected value is a weighted average of all possible values. In other words, each possible value that the random variable can assume is multiplied by its assigned weight, and the resulting products are then added together to find the expected value.
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=the number of heads in two coin tosses
Possible outcome
Probability mass function (p.m.f.)
,
Mean of
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Variance of a Random Variable
Indicates how far the outcomes of a random variable are spread out. A non-zero variance is always positive:
a small variance indicates that the outcomes tend to be very close to the mean and hence to each other;
a high variance indicates that the outcomes are very spread out from the mean and from each other.
The square root of variance is called the standard deviation.
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Normal random variable,
Continuous random variable takes values in
Defined by two parameters:
Mean , Variance
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Basic Statistics
Finding mean and variance of a series of data
Using histogram to denote frequency
See Normal RV.xlsm, worksheet “Histogram”
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How Z chart is used
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Year No of costumes sold
2010195
2011200
2012200
2013210
2014205
Average Costumes per year 202
STDEV5.700877125
Total service level 0.9
z ( from the Table for 0.90)1.185
Order quantity for the year 2015 = AVG + STDEV*z208.7555394
Round number209
Sample calculation for order quantity
Sheet1
| Sample calculation for order quantity | |||||
| Year | No of costumes sold | ||||
| 2010 | 195 | 0.5530351166 | |||
| 2011 | 200 | ||||
| 2012 | 200 | ||||
| 2013 | 210 | ||||
| 2014 | 205 | ||||
| Average Costumes per year | 202 | ||||
| STDEV | 5.7008771255 | ||||
| Total service level | 0.9 | ||||
| z ( from the Table for 0.90) | 1.185 | ||||
| Order quantity for the year 2015 = AVG + STDEV*z | 208.7555393937 | ||||
| Round number | 209 |