Chapter 1 Fundamentals of Applied Probability by Al Drake.
-
Upload
roxanne-lawson -
Category
Documents
-
view
238 -
download
2
Transcript of Chapter 1 Fundamentals of Applied Probability by Al Drake.
![Page 1: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/1.jpg)
Chapter 1
Fundamentals of Applied
Probability by Al Drake
![Page 2: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/2.jpg)
Experiment Any non-deterministic process. For example:
Model of an Experiment A simplified description of an experiment. For example: - the number on the top face
![Page 3: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/3.jpg)
Sample Space All possible outcomes of an experiment.
“The precise statement of an appropriate sample space, resulting from the detailed description of a model of an experiment, will do much to resolve common difficulties [when solving problems].
In this book we shall literally live in sample space.” (pg .6)
![Page 4: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/4.jpg)
Visualizing the Sample Space
(1)(2) (3) (4) (5)
(6)
sample points (S1, S2, etc.)
universe (U)
![Page 5: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/5.jpg)
Different ways of listingexperimental outcomes:
(1)(2) (3) (4) (5)
(6)
(divisible by 2) (odd)(even)
(divisible by 3)
(5)
![Page 6: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/6.jpg)
Sample Space All possible outcomes of an experiment.
Sample points must be: - finest grain no 2 distinguishable outcomes share a point
- mutually exclusive no outcome maps to more than 1 point
- collectively exhaustive all possible outcomes map to some point
O1
O2
S1
S1
S2O1
??O1
![Page 7: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/7.jpg)
Different ways of listingexperimental outcomes:
(1)(2) (3) (4) (5)
(6)
(divisible by 2) (odd)(even)
(divisible by 3)
(5)
![Page 8: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/8.jpg)
Event Defined on a sample space. A grouping of 1 or more sample points.
(1)
(2)
(3)
(4)
(5)
(6)
{even}{< 3}
{5}
Warning: This is subtly unintuitive.
![Page 9: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/9.jpg)
Event Space A set of 1 or more events that covers all outcomes of an experiment.
![Page 10: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/10.jpg)
Different ways of definingevents:
(1)
(2)
(3)
(4)
(5)
(6)
(1)
(2)
(3)
(4)
(5)
(6)
(1)
(2)
(3)
(4)
(5)
(6)
{even}
{odd}
{2} {4} {6}
{1} {3} {5}
{5}
{divisible by 2}
{divisible by 3}
![Page 11: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/11.jpg)
Sample Event Space A set of 1 or more events that covers all outcomes of an experiment.
Event points must be: - mutually exclusive no outcome maps to more than 1 event
- collectively exhaustive all possible outcomes map to some event
![Page 12: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/12.jpg)
Different ways of definingevents:
(1)
(2)
(3)
(4)
(5)
(6)
(1)
(2)
(3)
(4)
(5)
(6)
(1)
(2)
(3)
(4)
(5)
(6)
{even}
{odd}
{2} {4} {6}
{1} {3} {5}
{5}
{divisible by 2}
{divisible by 3}
![Page 13: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/13.jpg)
Operations on events
A = {< 3}
(1)
(2)
(3)
(4)
(5)
(6)
A’
complement
![Page 14: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/14.jpg)
Operations on events
A = {< 3}
(1)
(2)
(3)
(4)
(5)
(6)
AB
intersection
B = {odd}
![Page 15: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/15.jpg)
Operations on events
A = {< 3}
(1)
(2)
(3)
(4)
(5)
(6)
A + B
union
B = {odd}
Venn Diagram – picture of events in the universal set
![Page 16: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/16.jpg)
Axioms
![Page 17: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/17.jpg)
Theorems
![Page 18: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/18.jpg)
Proving a Theorem: CD = DC
Start with axiom 1: A+B = B+A
Warning: A + B is not addition, AB is not multiplication A + B = A + B + C does not mean C = φ
A B
C
![Page 19: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/19.jpg)
How do you establish the sample space?
Sequential Coordinate System
![Page 20: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/20.jpg)
How do you establish the sample space?
![Page 21: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/21.jpg)
Probability A number assigned to an event which represents the relative likelihood that event will occur when the experiment is performed.
For event A, its probability is P(A)
![Page 22: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/22.jpg)
Axioms of Probability
![Page 23: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/23.jpg)
Theorems
Warning: There are 2 types of + operator here
![Page 24: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/24.jpg)
Conditional Probability
![Page 25: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/25.jpg)
Conditional Probability
A
B
shift to conditional probability space
B
A
A
B
0.1 0.4 0
0.30.1
0.1
P(A|B) ?
![Page 26: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/26.jpg)
Sequential sample spaces revisited
![Page 27: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/27.jpg)
Independence Two events A and B, defined on a sample space, are independent if knowledge as to whether the experimental outcome had attribute A would not affect our measure of the likelihood that the experimental outcome also had attribute B.
Formally: P(A|B) = P(A)
P(AB) = P(A)*P(B)
NOTE: A may be φ
![Page 28: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/28.jpg)
Mutual Independence
For example, if you have 5 events defined, any subset of these events has the property: P(A1 A2 A5) = P(A1)P(A2)P(A5)
![Page 29: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/29.jpg)
Mutual Independence Pair-wise independence doesn’t meanmutual independence.
If we know:P(A1 A2) = P(A1)P(A2)P(A1 A3) = P(A1)P(A3)P(A2 A3) = P(A1)P(A3)
This does not imply mutual independence:P(A1 A2 A3) = P(A1)P(A2)P(A3)
![Page 30: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/30.jpg)
Conditional Independence
![Page 31: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/31.jpg)
Bayes Theorem
If a set of events A1, A2, A3… form an event space:
![Page 32: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/32.jpg)
Permutations, Combinations
![Page 33: Chapter 1 Fundamentals of Applied Probability by Al Drake.](https://reader035.fdocuments.in/reader035/viewer/2022081506/56649ef05503460f94c0048b/html5/thumbnails/33.jpg)
Bonus Problem
1 2 331