summary -1 chapters 2-6 of decoursey. basic probability (chapter 2, w.j.decoursey, 2003) objectives:...
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Summary -1
Chapters 2-6 of DeCoursey
Basic Probability(Chapter 2, W.J.Decoursey, 2003)
Objectives:
- Define probability and its relationship to relative frequency of an event.
- Learn the basic rules of combining probabilities.
- Understand the concepts of mutually exclusive / not mutually exclusive and independent / not independent events.
- Apply these concepts to solve sample problems.
Basic ProbabilityBasic Rules of Combining Probability:
- Addition Rule: Pr [event A] = C, Pr [event B] = D, what is the probability of
Pr [event A or event B] ? Equals C+D?
Case one: Mutually exclusive events: if one event occurs, other events can not occur. There is no overlap. The probability of occurrence of one or another of more than one event is the sum of the probabilities of the separate events.
Example one.A B Venn Diagram
Pr [ A U B ] = Pr [A] +Pr [B]Pr [ A U B ] = Pr [occurrence of A or B or Both]
Basic ProbabilityBasic Rules of Combining Probability:
- Addition Rule:Case two: Not mutually exclusive events: there can be
overlap between them. The probability of overlap must be subtracted from the sum of probabilities of the separate events.
A BA∩B
Pr [ A U B ] = Pr [occurrence of A or B or Both]
Pr [ A U B ] = Pr [A] +Pr [B] – Pr [A ∩ B]
Pr [A ∩ B] = Pr [occurrence of both A and B]
Venn Diagram
Basic ProbabilityBasic Rules of Combining Probability:
- Multiplication Rule:a. The basic idea for calculating the number of choices:
- There are n1 possible results from one operation.
- For each of these, there are n2 possible results from a second operation.
- Then here are (n1Xn2) possible outcomes of the two operations together.
Basic ProbabilityBasic Rules of Combining Probability:
- Multiplication Rule:b. Independent events
The occurrence of one event does not affect the probability of the occurrence of another event.
The probability of the individual events are multiplied to give the probability of them occurring together.
Consider 2 events, A and B. Then the probability of A and B occurring together is
Pr [A] * Pr [B]
Note the use of logical AND
Basic ProbabilityBasic Rules of Combining Probability:
- Multiplication Rule:c. Not-independent events
The occurrence of one event affects the probability of the occurrence of another event.
The probability of the affected event is called the conditional probability since it is conditional upon the first event taking place.
The multiplication rule then becomes for A and B occurring together
Pr [A ∩ B]=Pr [A] * Pr [B|A]
Pr [B|A] : conditional probability of B. (examples in class)
Basic ProbabilitySummary of Combining Rule
When doing combined probability problems, ask yourself:
1. Does the problem ask the logical OR or the logical AND?
2. If OR, ask your self are the events mutually exclusive or not? If yes, Pr [ A U B ] = Pr [A] +Pr [B],
other wise
Pr [ A U B ] = Pr [A] +Pr [B] – Pr [A ∩ B]
3. If AND, use the multiplication rule and remember conditional probability. A probability tree may be helpful.
4. Fault Tree Analysis
Basic ProbabilityPermutations and Combinations:- Permutations:
a. A total of n distinguishable items to be arranged. R items are chosen at a time (r ≤ n). The number of permutations of n items chosen r at a time is written nPr.
(n-r)!
n!
rnnnn
)1)...(2)(1)((Prn
Basic ProbabilityPermutations and Combinations:- Permutations:
b. To calculate the number of permutations into class.
A total of n items to be placed. n1 items are the same of one class, n2 are the same of the second class and n3 are the same as a third class.
n1+n2+n3=n
The number of permutations of n items taken n at a time:
!nnn
n!
321rn !!
P
Basic Probability
Permutations and Combinations:
- Combinations: c. Similar to Permutations but taking no account of order.
The number of combinations of n items taken r at a time:
r!rn
n!Crn )!(r!
Prn
Descriptive Statistics
Objectives: (Chapter 3, Decoursey)
- To understand the definition of mean, median, variance, standard deviation, mean absolute deviation and coefficient variation and calculate these quantities.
- To calculate some of these quantities using the statistical functions of Excel.
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Descriptive Statistics
Probability Distributions, Discrete Random Variables
Objectives: (Chapter 5, DeCoursey)
- To define a probability function, cumulative probability, probability distribution function and cumulative distribution functions.
- To define expectation and variance of a random variable.
- To determine probabilities by using Binomial Distribution.
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Probability Distributions, Discrete Random Variables
Binomial Distribution: Let p = probability of “success”
q=probability of “failure” = 1-p
n = number of trails
r = number of “success” in “n” trails
Then the probability of r successes for n trials is given by the following general formula:
Probability Distributions, Discrete Random Variables
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Probability Distributions, Continuous Variables
Objectives: (Chapter 6, DeCoursey)
- To establish the difference between probability distribution for discrete and continuous variables.
- To learn how to calculate the probability that a random variable, X, will fall between the limits of “a” and “b”.
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Continuous Variable Discrete Variable