chapter 2: probability concepts and applications...2 learning objectives after completing this...
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Chapter 2: Probability Concepts and
Applications
Textbook: pp. 39-80
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Learning Objectives
After completing this chapter, students will be able to:
• Understand the basic foundations of probability analysis.
• Describe statistically dependent and independent events.
• Use Bayes’ theorem to establish posterior probabilities.
• Describe and provide examples of both discrete and
continuous random variables.
• Explain the difference between discrete and continuous
probability distributions.
• Calculate expected values and variances and use the
normal table.
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Learning Objectives
After completing this chapter, students will be able to:
• Understand the binomial distribution.
• Understand the normal distribution and use the normal
table.
• Understand the F distribution.
• Understand the exponential distribution and its relation to
queuing theory.
• Understand the Poisson distribution and its relation to
queuing theory.
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• Life is uncertain; we are not sure what the future will
bring
• Probability is a numerical statement about the
likelihood that an event will occur
Introduction
For example, the market for the new Iphone X might be
“good” with a chance of 60% (a probability of 0.6) or “not
good” with a chance of 40% (a probability of 0.4).
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• The probability, P, of any event or state of nature
occurring is greater than or equal to 0 and less than or
equal to 1. That is,
0 ≤ P(event) ≤ 1
• A probability of 0 indicates that an event is never
expected to occur. A probability of 1 means that an
event is always expected to occur.
• The sum of the simple probabilities for all possible
outcomes of an activity must equal 1. Regardless of
how probabilities are determined, they must adhere to
these two rules.
Two Basic Rules of Probability
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• Objective Approach
• Relative frequency approach
Types of Probability (1 of 3)
P(event) =Number of occurrences of the event
Total numbers of trials or outcomes
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• Historical demand for white latex paint at = 0, 1, 2, 3, or
4 gallons per day
• Observed frequencies over the past 200 days
Diversey Paint Example (1 of 2)
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• Historical demand for white latex paint at = 0, 1, 2, 3, or
4 gallons per day
• Observed frequencies over the past 200 days
Diversey Paint Example (2 of 2)
Individual
probabilities are all
between 0 and 1
0 ≤ P (event) ≤ 1
Total of all event
probabilities equals 1
∑ P (event) = 1.00
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• Objective Approach
o Classical or logical method
• Perform a series of trials
Types of Probability (2 of 3)
1head =
2
13spade =
52
01
= .25=4
= 25%
Number of ways of getting a headP
Number of possible outcomes head or tail
Number of chances of drawing a spadeP
Number of possible outcomes
( )
2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace
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• Subjective Approach
o When logic and past history are not appropriate,
probability values can be assessed subjectively!
o Based on the experience and judgment of the
person making the estimate
• Opinion polls (e.g. to determine election returns)
• Judgment of experts
• Delphi method (a panel of experts is assembled to
make their predictions of the future)
Types of Probability (3 of 3)
Example: What is the probability that the Chinese
economy will be in a severe depression in 2018?
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• Events are said to be mutually exclusive if only one of
the events can occur on any one trial
o Tossing a coin will result in either a head or a tail
o Rolling a die will result in only one of six possible
outcomes
Mutually Exclusive and Collectively
Exhaustive Events (1 of 2)
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• Events are said to be
collectively exhaustive if
the list of outcomes
includes every possible
outcome
• Both heads and tails as
possible outcomes of coin
flips
• All six possible outcomes of
the roll of a die:
Mutually Exclusive and Collectively
Exhaustive Events (2 of 2)
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Venn Diagrams
A B
Events that are mutually exclusive
Events that are notmutually exclusive
A B
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• Draw one card from a deck of 52 playing cards
A = event that a 7 is drawn
B = event that a heart is drawn
P (a 7 is drawn) = P(A)= 4/52 = 1/13
P (a heart is drawn) = P(B) = 13/52 = 1/4
• These two events are not mutually exclusive since a 7
of hearts can be drawn
• These two events are not collectively exhaustive
since there are other cards in the deck besides 7s and
hearts
Drawing a Card
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• Intersection – the set of all outcomes that are common to
both events
• Intersection of event A and event B = A and B
= A ∩ B
= AB
• Probability notation
P(Intersection of event A and event B) = P(A and B)
= P(A ∩ B)
= P(AB)
Sometimes called joint probability
Unions and Intersections of Events (1 of 3)
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• Union – the set of all outcomes that are contained in
either of two events
Union of event A and event B = A or B
• Probability notation
P(Union of event A and event B) = P(A or B)
= P(A ∪ B)
Unions and Intersections of Events (2 of 3)
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• In the previous example
• Intersection of event A and event B
(A and B) = the 7 of hearts is drawn
P(A and B) = P(7 of hearts is drawn) = 1/52
• Union of event A and event B
(A or B) = either a 7 or a heart is drawn
P(A or B) = P(any 7 or any heart is drawn) = 16
/52
Unions and Intersections of Events (3 of 3)
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• General rule for union of two events,
additive rule
P(A or B) = P(A) + P(B) – P(A and B)
o Union of two events, a 7 or a heart
P(A or B) = P(A) + P(B) – P(A and B)
= 4/52 + 13/52 – 1/52
= 16/52
Probability Rules (1 of 5)
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• Conditional probability – probability that an event
occurs given another event has already happened
o Probability of a 7 given a heart has been drawn
Probability Rules (2 of 5)
P(A | B) = = = 1/13
P(AB)
P(B)
1/52
13/52
P(A | B) =P(AB)
P(B)
P(AB) = P(A | B) P(B)
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• Independent– one event has no effect on the other event
P(A | B) = P(A)
P(A and B) = P(A)P(B)
• For a fair coin tossed twice
A = event that a head is the result of the first toss
B = event that a head is the result of the second toss
P(A) = 0.5 and P(B) = 0.5
P(AB) = P(A)P(B) = 0.5(0.5) = 0.25
Probability Rules (5 of 5)
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• A bucket contains 3 black balls and 7 green balls
Draw a ball from the bucket, replace it, and draw a
second ball!
1. The probability of a black ball drawn on first draw is:
P(B) = 0.30
2. The probability of two green balls drawn is:P(GG) = P(G) x P(G) = 0.7 x 0.7 = 0.49
Independent Events (1 of 2)
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• A bucket contains 3 black balls and 7 green balls
Draw a ball from the bucket, replace it, and draw a
second ball!
3. The probability of a black ball drawn on the second
draw if the first draw is green is:
P(B | G) = P(B) = 0.30
4. The probability of a green ball drawn on the second
draw if the first draw is green is:
P(G | G) = P(G) = 0.70
Independent Events (2 of 2)
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• An urn contains the following 10 balls:
o 4 are white (W) and lettered (L)
o 2 are white (W) and numbered (N)
o 3 are yellow (Y) and lettered (L)
o 1 is yellow (Y) and numbered (N)
P(WL) = 4/10 = 0.4 P(YL) = 3/10 = 0.3
P(WN) = 2/10 = 0.2 P(YN) = 1/10 = 0.1
P(W) = 6/10 = 0.6 P(L) = 7/10 = 0.7
P(Y) = 4/10 = 0.4 P(N) = 3/10 = 0.3
Dependent Events (1 of 3)
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Dependent Events (2 of 3)
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• The conditional probability that the ball drawn is
lettered, given that it is yellow
We can verify P(YL) using the joint probability formula:
Dependent Events (3 of 3)
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Which sets are independent?1. (a) Your education
(b) Your income level
2. (a) Draw a jack of hearts from a full 52-card deck
(b) Draw a jack of clubs from a full 52-card deck
3. (a) Snow in Zhengzhou
(b) Rain in Bangor / Wales
Probability Rules (3 of 5)
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Which sets are independent?1. (a) Your education
(b) Your income level
2. (a) Draw a jack of hearts from a full 52-card deck
(b) Draw a jack of clubs from a full 52-card deck
3. (a) Snow in Zhengzhou
(b) Rain in Bangor / Wales
Probability Rules (4 of 5)
Dependent events
Independent events
Independent events
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• Bayes’ theorem is used to incorporate additional
information and help create posterior probabilities
from original or prior probabilities
Revising Probabilities with Bayes’ Theorem (1 of 7)
This means that we can take new or recent
data and then revise and improve upon our
old probability estimates for an event!
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• A cup contains two dice identical in appearance but one
is fair (unbiased) and the other is loaded (biased)
o The probability of rolling a 3 on the fair die is 1/6 or 0.166
o The probability of tossing the same number on the loaded
die is 0.60
o We select one by chance, toss it, and get a 3
o What is the probability that the die rolled was fair?
o What is the probability that the loaded die was rolled?
Revising Probabilities with Bayes’ Theorem (2 of 7)
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• The probability of the die being fair or loaded is
P(fair) = 0.50 P(loaded) = 0.50
and that
P(3 | fair) = 0.166 P(3 | loaded) = 0.60
• The probabilities of P(3 and fair) and P(3 and loaded)
are
P(3 and fair) = P(3 | fair) x P(fair) = (0.166)(0.50) = 0.083
P(3 and loaded) = P(3 | loaded) x P(loaded) = (0.60)(0.50)
= 0.300
Revising Probabilities with Bayes’ Theorem (3 of 7)
We can answer these questions by using the formula for joint
probability under statistical dependence and Bayes’ theorem.
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• A 3 can occur in combination with the state “fair die” or
in combination with the state “loaded die”.
• The sum of their probabilities gives the unconditional
(marginal probability) of a 3 on the toss:
P(3) = 0.083 + 0.300 = 0.383
Revising Probabilities with Bayes’ Theorem (4 of 7)
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• If a 3 does occur, the probability that the die rolled was
the fair one is
• The probability that the die was loaded is
Revising Probabilities with Bayes’ Theorem (5 of 7)
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• If a 3 does occur, the probability that the die rolled was
the fair one is
• The probability that the die was loaded is
• These are the revised or posterior probabilities for the
next roll of the die
• We use these to revise our prior probability estimates
Revising Probabilities with Bayes’ Theorem (6 of 7)
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• We can compute revised probabilities more directly by
using
where
A’ = the complement of the event A;
for example, if A is the event “fair die”, then A’ is “loaded
die”
General Form of Bayes’ Theorem (1 of 2)
P(A | B) =P(B | A)P(A)
P(B | A)P(A)+P(B | ¢A )P( ¢A )
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• Conditional probability
From the previous example
Replace A with “fair die”, A’ with “loaded die”, B with “3
rolled”
General Form of Bayes’ Theorem (2 of 2)
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• Please read Chapter 3!
Homework --- Chapter 2