Transcript Basic Rules of Combining Probability

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 Probability
Basic 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
Pr [ A U B ] = Pr [A] +Pr [B]
Pr [ A U B ] = Pr [occurrence of A or B or Both]
B
Venn Diagram
Basic Probability
Basic 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
A∩B
B
Pr [ A U B ] = Pr [A] +Pr [B] – Pr [A ∩ B]
Pr [ A U B ] = Pr [occurrence of A or B or Both]
Pr [A ∩ B] = Pr [occurrence of both A and B]
Venn Diagram
Basic Probability
Basic 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 Probability
Basic 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 Probability
Basic 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 Probability
Summary 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 Probability
Permutations 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
Pr  (n)( n  1)( n  2)...( n  r  1)
n!

(n-r)!
Basic Probability
Permutations 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:
n!
n Pr 
n1!n 2 ! n3!
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:
Pr
n!

n Cr 
r!
(n  r )! r!
n
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.
Descriptive Statistics
1
x or  
N
N
 
2
 (x
i 1
i
x
/ or s/ x
i
 )
N
2
s 
2
N
N

2
(
x


)
 i
i 1
N
 (x
i 1
i
 x)
N 1
N
s
2
(
x

x
)
 i
i 1
N 1
2
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.
Probability Distributions, Discrete Random
Variables
k
Pr[ X  x]   p( xi )
 p( x )  1
p ( xi )  0
i
i 0
xi  x
p( xi )  Pr[ X  xi ]  Pr[ X  xi 1 ]
E( X )   x 
 ( x ) p( x )
i
i
all xi
 x  E( X   x )  E[ X ]  
2
2
2
where E[ X ]   x p( xi )
2
2
x
2
i
all i
 x  E[( X   x ) ]  E ( X )   x
2
2
2
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:
n!
Pr[ R  r ] 
p r q ( nr )
r!(n  r )!
 n C r p r q ( nr )
E ( R)    np
  npq
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”.
Continuous Variable
Discrete Variable
 p( x )
b
Pr[ a  x  b]   f ( x)dx
a  xi b
a
x1
Pr[ X  x1 ] 

f ( x)dx
Pr[ X  x]   p( xi )
xi  x

 p( x )  1

F ( ) 

f ( x)dx  1

  E( X ) 
i
all xi

 xf ( x)dx
E( X )   x 
x 
2
x

2
f ( x)dx   x
2
 ( x ) p( x )
i
i
all xi


i
 x   xi2 p( xi )   x2
2
all i