1. Theory of Probability
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2. Randomness The term randomness suggests unpredictability
A simple example of randomness is the tossing of a coin. The outcome is uncertain; it can either be an observed head (H) or an observed tail (T). Because the outcome of the toss cannot be predicted for sure, we say it displays randomness.
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3. Uncertainty
At some time or another, everyone will experience uncertainty. For example, you are approaching the traffic signals and the light changes from green to amber. You have to decide whether you can make it trough the intersection or not. You may be uncertain as to what the correct decision should be.
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4. Probability
The concept of probability is used to quantify this measure of doubt. If you believe that you have a 0.99 probability of getting across the intersection, you have made a clear statement about your doubt. The probability statement provides a great deal of information, much more than statements such as “Maybe I can make it across” or “I should make it across”
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5. Important Definitions
Random Experiment – a process leading to an uncertain outcome
Basic Outcome – a possible outcome of a random experiment
Sample Space – the collection of all possible outcomes of a random experiment
Event – any subset of basic outcomes from the sample space
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6. Important Definitions
(continued)
Intersection of Events – If A and B are two events in a sample space S, then the intersection, A ∩ B, is the set of all outcomes in S that belong to both A and B S A
AB B
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7. Important Definitions
(continued)
A and B are Mutually Exclusive Events if they have no basic outcomes in common
i.e., the set A ∩ B is empty S A B
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8. Important Definitions
(continued)
Union of Events – If A and B are two events in a sample space S, then the union, A U B, is the set of all outcomes in S that belong to either
A or B S
The entire shaded area represents
A U B A B
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9. Important Definitions
(continued)
The Complement of an event A, , is the set of all basic outcomes in the sample space that do not belong to A. S A
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10. Examples
Let the Sample Space be the collection of all possible outcomes of rolling one dice:
S = [1, 2, 3, 4, 5, 6]
Let A be the event “Number rolled is even”
Let B be the event “Number rolled is at least 4”
Then
A = [2, 4, 6] and B = [4, 5, 6]
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11. Examples S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]
(continued)
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]
Complements:
Intersections:
Unions:
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12. Examples Mutually exclusive: Collectively exhaustive:
(continued)
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]
Mutually exclusive:
A and B are not mutually exclusive
The outcomes 4 and 6 are common to both
Collectively exhaustive:
A and B are not collectively exhaustive
A U B does not contain 1 or 3
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13. Probability
Probability – the chance that an uncertain event will occur
Mathematically, any function P(.) with
0 ≤ P(A) ≤ 1
P(ø)=0, P(S)=1 1
Certain .5
Impossible
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14. Assessing Probability
There are three approaches to assessing the probability of an uncertain event:
1. classical probability
Assumes all outcomes in the sample space are equally likely to occur
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15. Counting the Possible Outcomes
Use the Combinations formula to determine the number of combinations of n things taken k at a time
where
n! = n(n-1)(n-2)…(1)
0! = 1 by definition
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16. Assessing Probability
Three approaches (continued)
2. relative frequency probability
the limit of the proportion of times that an event A occurs in a large number of trials, n
3. subjective probability
an individual opinion or belief about the probability of occurrence
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

17. Probability Distributions
Random Variable
Represents a possible numerical value from a random experiment
Random
Variables
Discrete
Random Variable
Continuous
Random Variable
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18. Probability Distributions
Discrete
Probability Distributions
Continuous
Probability Distributions
Binomial
Uniform
Hypergeometric
Normal
Poisson
Exponential
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19. Discrete Random Variables
Can only take on a countable number of values
Examples:
Roll a dice twice
Let X be the number of times 4 comes up
(then X could be 0, 1, or 2 times)
Toss a coin 5 times.
Let X be the number of heads
(then X could be 0, 1, 2, 3, 4, or 5)
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20. Discrete Probability Distribution
Experiment: Toss 2 Coins. Let X = # heads.
Show P(x) , i.e., P(X = x) , for all values of x:
4 possible outcomes
Probability Distribution T T
x Value Probability
/4 = .25
/4 = .50
/4 = .25 T H H T
.50
.25
Probability H H x
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21. The Normal Distribution
‘Bell Shaped’
Symmetrical
Mean, Median and Mode
are Equal
Location is determined by the mean, μ
Spread is determined by the standard deviation, σ
The random variable has an infinite theoretical range:
+  to  
f(x) σ x μ
Mean
= Median
= Mode
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22. The Normal Distribution
(continued)
The normal distribution closely approximates the probability distributions of a wide range of random variables
Distributions of sample means approach a normal distribution given a “large” sample size
Computations of probabilities are direct and elegant
The normal probability distribution has led to good business decisions for a number of applications
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23. Many Normal Distributions
By varying the parameters μ and σ, we obtain different normal distributions
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24. The Normal Distribution Shape
f(x)
Changing μ shifts the distribution left or right.
Changing σ increases or decreases the spread. σ μ x
Given the mean μ and variance σ we define the normal distribution using the notation
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25. The Normal Probability Density Function
The formula for the normal probability density function is
Where e = the mathematical constant approximated by
π = the mathematical constant approximated by
μ = the population mean
σ = the population standard deviation
x = any value of the continuous variable,  < x < 
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26. Cumulative distribution function - CDF
Also called probability distribution function or just distribution function
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27. Cumulative Probability Function
The cumulative probability function, denoted F(x0), shows the probability that X is less than or equal to x0
In other words,
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28. Cumulative Normal Distribution
For a normal random variable X with mean μ and variance σ2 , i.e., X~N(μ, σ2), the cumulative distribution function is
f(x) x0 x
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29. CDF vs. pdf
Cumulative distribution function is an integral of probability density function
Probability density function is a derivation of cumulative distribution function.
CDF is defined only for values of a random variable X that are less than a certain value (x). CDF represents the surface under the probability density function.
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30. Finding Normal Probabilities
The probability for a range of values is measured by the area under the curve x a μ b
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31. Finding Normal Probabilities
(continued) x a μ b x a μ b x a μ b
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32. Upper Tail Probabilities
Suppose X is normal with mean 8.0 and standard deviation 5.0.
Now Find P(X > 8.6) X
8.0
8.6
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33. The Standardized Normal
Any normal distribution (with any mean and variance combination) can be transformed into the standardized normal distribution (Z), with mean 0 and variance 1
Need to transform X units into Z units by subtracting the mean of X and dividing by its standard deviation
f(Z) 1 Z
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34. Example
If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is
This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.
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35. Comparing X and Z units 100 200 X 2.0 Z
(μ = 100, σ = 50)
2.0 Z
(μ = 0, σ = 1)
Note that the distribution is the same, only the scale has changed. We can express the problem in original units (X) or in standardized units (Z)
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36. Finding the X value for a Known Probability
Steps to find the X value for a known probability:
1. Find the Z value for the known probability
2. Convert to X units using the formula:
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37. Finding the X value for a known probability
(continued)
Example:
Suppose X is normal with mean 8.0 and standard deviation 5.0.
Now find the X value so that only 20% of all values are below this X
.2000 X ?
8.0 Z ?
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38. Find the Z value for 20% in the Lower Tail
1. Find the Z value for the known probability
Standardized Normal Probability
Table (Portion)
20% area in the lower tail is consistent with a Z value of -0.84 z
F(z)
.82
.7939
.80
.20
.83
.7967
.84
.7995 X ?
8.0
.85
.8023 Z
-0.84
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39. Finding the X value 2. Convert to X units using the formula:
So 20% of the values from a distribution with mean 8.0 and standard deviation 5.0 are less than 3.80
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40. The Empirical Rule
If the data distribution is bell-shaped, then the interval:
contains about 68% of the values in the population or the sample
68%
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41. The Empirical Rule contains about 95% of the values in
the population or the sample
contains about 99.7% of the values in the population or the sample
95%
99.7%
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42. Next topic
Point and interval estimate
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