![Page 1: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/1.jpg)
Chapter Two
Probability
![Page 2: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/2.jpg)
Probability Definitions
Experiment:Process that generates
observations.
Sample Space:Set of all possible
outcomes of an experiment.
![Page 3: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/3.jpg)
![Page 4: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/4.jpg)
Event Definitions Event: Subset of outcomes contained in the sample space.
Simple Event: Consists of exactly one outcome.
Compound Event: Consists of more than one outcome.
![Page 5: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/5.jpg)
Set Notation ReviewFor Two Events A and B:
Union: “A or B” = A B
Intersection: “A and B” = A B
Complement: A´Mutually Exclusive:
No outcomes in common
![Page 6: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/6.jpg)
![Page 7: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/7.jpg)
Probabilistic Models1) Equally Likely:
Based on Definition Games of Chance
2) Relative Frequency Objective Interpretation Based on Empirical Data
3) Personal Probability Subjective
Interpretation Based on Degree of
Belief
![Page 8: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/8.jpg)
Properties of ProbabilityFor any Event A:
P(A) = 1 – P(A)
If A and B are Mutually Exclusive, P(A B) = 0
For any two events A and B:P(A B) = P(A) + P(B) – P(A B)
![Page 9: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/9.jpg)
Counting Techniques
Product Rule for Ordered PairsTree DiagramsGeneral Product Rule PermutationsCombinations
![Page 10: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/10.jpg)
![Page 11: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/11.jpg)
PermutationAn “ordered” arrangement ofk distinct objects taken from a set of n distinct objects.
The number of ways of ordering n distinct objects taken k at a time is Pk,n
Pk,n = n! / (n-k)!
![Page 12: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/12.jpg)
![Page 13: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/13.jpg)
Combination An “unordered” arrangement of k distinct objects taken from a set of n distinct objects.
The number of ways of ordering n distinct objects taken k at a time is Ck,n
Ck,n = (nk) = n! / k!(n-k)!
![Page 14: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/14.jpg)
Example:Twenty Five tickets are sold in a lottery, with the first, second, and third prizes to be determined by a random drawing. Find the number of different ways of drawing the three winning tickets.
![Page 15: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/15.jpg)
Example:Twenty tickets are sold in a lottery, with 5 round trips to game 1 of the World Series to be determined by a random drawing. Find the number of different ways of drawing the five winning tickets.
![Page 16: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/16.jpg)
Example: A solar system contains 6 Earth-like planets & 4 Gas Giant-like planets. How many ways may we explore this solar system if our resources allow us to only probe 3 Gas Giants and 3 Earth-like planets?
![Page 17: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/17.jpg)
Example:There are 50 students in ISE 261. What is the probability that at least 2 students have the same birthday? (Ignore leap years).
![Page 18: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/18.jpg)
ExampleA dispute has risen in Watson Engineering concerning the alleged unequal distribution of 10 computers to three different engineering labs. The first lab (considered to be abominable) required 4 computers; the second lab and third lab needed 3 each. The dispute arose over an alleged ISE 261 random distribution of the computers to the labs which placed all 4 of the fastest computers to the first lab. The Dean desires to known the number of ways of assigning the 10 computers to the three labs before deciding on a course of action. What is the Dean’s next question?
![Page 19: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/19.jpg)
Conditional ProbabilityFor any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by: P(A|B) = P(A B)/P(B)
![Page 20: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/20.jpg)
![Page 21: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/21.jpg)
![Page 22: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/22.jpg)
Multiplication Rule
P(A B) = P(A|B) x P(B)
![Page 23: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/23.jpg)
Multiplication RuleFour students have responded to a request by a blood bank. Blood types of each student are unknown. Blood type A+ is only needed. Assuming one student has this blood type; what is the probability that at least 3 students must be typed to obtain A+?
![Page 24: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/24.jpg)
Conditional ProbabilityExperiment = One toss of a coin.If the coin is Heads; one die is thrown. Record Number.If the coin is Tails; two die are thrown. Record Sum.
What is the Probability that the recorded number will equal 2?
![Page 25: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/25.jpg)
Conditional Probability Problem:30% of interstate highway accidents involve alcohol use by at least one driver (Event A). If alcohol is involved there is a 60% chance that excessive speed (Event S) is also involved; otherwise, this probability is only 10%. An accident occurs involving speeding! What is the probability that alcohol is involved?
P(A) = .30 P(SA|A) = .60
P(A’)= .70 P(SA’|A’)= .10
![Page 26: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/26.jpg)
![Page 27: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/27.jpg)
Bayes’ Theorem
A1,A2,….,Ak a collection of k mutually exclusive and exhaustive events with P(Ai) > 0 for i = 1,…,k. For any other event B for which P(B) > 0:
P(Ap|B) = P (Ap B) / P(B) =
P(B|Ap) P(Ap) P(B|Ai) P(Ai)
![Page 28: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/28.jpg)
Example: Bayes’ TheoremThe probabilities are equal that any of 3 urns A1, A2,& A3 will be selected. Given an urn has been selected & the drawn ball is black; what is the probability that the selected urn was A3? A1 contains: 4 W & 1 Black A2 contains: 3 W & 2 Black A3 contains: 1 W & 4 Black
![Page 29: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/29.jpg)
IndependenceTwo events A and B are independent if:
P(A|B) = P(A)Or
P(B|A) = P(B)Or
P(A B) = P(A) P(B)and are dependent otherwise.
![Page 30: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/30.jpg)
Independence Example:Three brands of coffee, X, Y,& Z are to be ranked according to taste by a judge. Define the following events as:
A: Brand X is preferred to YB: Brand X is ranked BestC: Brand X is ranked SecondD: Brand X is ranked Third
If the judge actually has no taste preference & thus randomly assigns ranks to the brands, is event A independent of events B, C, & D?
![Page 31: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/31.jpg)
IndependenceConsider the following 3 events in the toss of a single die:A: Observe an odd numberB: Observe an even numberC: Observe an 1 or 2
Are A & B independent events?Are A & C independent events?
![Page 32: Chapter Two Probability. Probability Definitions Experiment: Process that generates observations. Sample Space: Set of all possible outcomes of an experiment](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d555503460f94a333ee/html5/thumbnails/32.jpg)
Example:A space probe to Mars has 35 electrical components in series. If the mission is to have a reliability (probability of success) of 0.90 & if all parts have the same reliability, what is the required reliability of each part?