section: housekeeping subsection: hygene

ProbabilityI will attempt to simulcast and record all lectures.
1 If you ask a question, you may be recorded.
2 If you feel sick, DO NOT COME TO CLASS.
1 Instead, follow remotely. 2 I will attempt to monitor the class chat.
3 If I feel sick, I will not come to class.
1 Watch your email for abrupt temporary moves to on-line teaching.
Housekeeping: Hygene Lecture 00 2 / 1
I will attempt to simulcast and record all lectures.
1 If you ask a question, you may be recorded. 2 If you feel sick, DO NOT COME TO CLASS.
1 Instead, follow remotely.
2 I will attempt to monitor the class chat.
3 If I feel sick, I will not come to class. 1 Watch your email for abrupt temporary moves to on-line teaching.
You must wear a mask in the classroom and in office hours.
1 This decision was taken by the Rutgers administration.
2 I am not authorized to modify it or grant exceptions. 3 Rutgers has committed to having masks in the classroom for your
use.
4 I will also be wearing a mask.
2 I am not authorized to modify it or grant exceptions.
3 Rutgers has committed to having masks in the classroom for your use.
1 Please use these resources responsibly.
use.
use. 1 Please use these resources responsibly.
use. 1 Please use these resources responsibly.
Section: Housekeeping
I will teach from slides.
1 Advantages:
1 Necessary to permit ill students to follow class. 2 Useful because my voice will be somewhat muffled wearing the mask. 3 Allows for faster covering of material.
2 Disadvantages
1 Preparation is very labor-intensive. 2 Allows for faster covering of material.
1 To counteract this effect, I will pose questions during class time and allow time for you to consider them before presenting a solution.
3 Breaks traditional note-taking
1 Note taking reinforces learning during class. 2 Note taking makes the process more interesting. 3 I encourage manual recording of key points of lectures.
4 Availability of substitutes for in-person participation may encourage class skipping.
Housekeeping: Class management Lecture 00 5 / 1
1 Advantages: 1 Necessary to permit ill students to follow class.
2 Useful because my voice will be somewhat muffled wearing the mask. 3 Allows for faster covering of material.
2 Disadvantages
1 Advantages: 1 Necessary to permit ill students to follow class. 2 Useful because my voice will be somewhat muffled wearing the mask.
3 Allows for faster covering of material.
2 Disadvantages
1 Advantages: 1 Necessary to permit ill students to follow class. 2 Useful because my voice will be somewhat muffled wearing the mask. 3 Allows for faster covering of material.
2 Disadvantages
2 Disadvantages
2 Disadvantages 1 Preparation is very labor-intensive.
2 Allows for faster covering of material.
2 Disadvantages 1 Preparation is very labor-intensive. 2 Allows for faster covering of material.
1 Note taking reinforces learning during class.
2 Note taking makes the process more interesting. 3 I encourage manual recording of key points of lectures.
1 Note taking reinforces learning during class. 2 Note taking makes the process more interesting.
3 I encourage manual recording of key points of lectures.
You will submit assignments on-line.
1 I will provide solutions.
2 Clear HW preparation is a courtesy to the grader.
3 See the course information sheet for grading policies.
Objectives Lecture 01
Section: Introduction:
Definition
1 Probability telling us how likely we are to see various outcomes of experiments or studies.
2 How often can refer to long run frequency, subjective assessment, etc.
3 May be a major theme in second semester.
Introduction:: Probability notions: Lecture 01 9 / 1
Definition
Definition
Examples:
1 How often will three heads arise in three tosses of a coin?
2 How often will a patient’s cancer go into remission after a certain type of chemotherapy?
3 How often will a certain company make investments of a certain level and dividends of a certain level?
Examples:
Examples:
Statistics is the study of inference about probabilistic mechanism generating data based on events observed.
1 Typical objectives: testing and estimation.
1 Ask yes–no questions about model: Hypothesis testing. 2 Pick a closest or collection of closest elements from a collection of
plausible alternative distns: Estimation. 3 Make a decision whose benefit depends on which probability distn is
operational: Decision Theory.
1 Typical objectives: testing and estimation. 1 Ask yes–no questions about model: Hypothesis testing.
2 Pick a closest or collection of closest elements from a collection of plausible alternative distns: Estimation.
3 Make a decision whose benefit depends on which probability distn is operational: Decision Theory.
Introduction:: Statistical Notions: Lecture 01 12 / 1
1 Typical objectives: testing and estimation. 1 Ask yes–no questions about model: Hypothesis testing. 2 Pick a closest or collection of closest elements from a collection of
plausible alternative distns: Estimation.
3 Make a decision whose benefit depends on which probability distn is operational: Decision Theory.
1 Typical objectives: testing and estimation. 1 Ask yes–no questions about model: Hypothesis testing. 2 Pick a closest or collection of closest elements from a collection of
plausible alternative distns: Estimation. 3 Make a decision whose benefit depends on which probability distn is
operational: Decision Theory.
Examples of statistical questions
1 Is it plausible that a coin giving heads and tails with equal frequency gave a total # of heads that we observe?
2 Is it plausible that a model of economic decision making in which dividends and investment are unrelated gave rise to an apparent relation between the two observed in the data?
3 Is it plausible that two treatments of equal utility gave rise to a difference in outcomes observed in a medical trial?
population vs. sample quantities:
1 We often are trying to describe or make decisions about a particular aspect of the probabilistic mechanism P generating the data;
1 these values are often called population values because they refer to the entire universe of outcomes from similar experiments.
2 such as the probability of seeing a head, 3 the difference in the probabilities of cures with different kinds of
treatments, or 4 a measure of association between two economic variables.
2 The experiment we observe often gives rise to analogous random measurements, called sample versions;
1 they refer to the outcome of the experiment we’ve actually performed, 2 and will likely be different if we rerun the experiment:
3 Examples
1 proportion of heads we actually see, 2 difference in the proportions of cures we actually see, 3 the association measured in the companies we look at.
3 Examples
2 such as the probability of seeing a head,
3 the difference in the probabilities of cures with different kinds of treatments, or
4 a measure of association between two economic variables.
3 Examples
treatments, or
4 a measure of association between two economic variables.
3 Examples
3 Examples
3 Examples
1 they refer to the outcome of the experiment we’ve actually performed,
2 and will likely be different if we rerun the experiment:
3 Examples
3 Examples
3 Examples
3 Examples 1 proportion of heads we actually see,
2 difference in the proportions of cures we actually see, 3 the association measured in the companies we look at.
3 Examples 1 proportion of heads we actually see, 2 difference in the proportions of cures we actually see,
3 the association measured in the companies we look at.
3 Examples 1 proportion of heads we actually see, 2 difference in the proportions of cures we actually see, 3 the association measured in the companies we look at.
Set notation is used to describe collections of outcomes
1 Sets are collections of outcomes.
1 A set of experimental outcomes are called an event. 2 Denote events using upper case non-bold Latin letters: A, B, C , ...
1 Script is based on medieval Latin. 2 Classical Latin did not have J, U, W .
2 The things contained in sets are called elements.
1 Denote elements by lower case non-bold Latin letters: a, b, c , ...
3 Let S represent the collection of all possible outcomes.
4 Mathematical quantities will be generally set in italic.
1 Sets are collections of outcomes. 1 A set of experimental outcomes are called an event.
2 Denote events using upper case non-bold Latin letters: A, B, C , ...
1 Sets are collections of outcomes. 1 A set of experimental outcomes are called an event. 2 Denote events using upper case non-bold Latin letters: A, B, C , ...
1 Script is based on medieval Latin.
2 Classical Latin did not have J, U, W .
2 The things contained in sets are called elements. 1 Denote elements by lower case non-bold Latin letters: a, b, c , ...
Continued
1 Let ∈ indicate that the element before the sign lies in the set after the sign:
1 a ∈ A, read a is an element of A. 2 If it is not true that a ∈ A, then we write a /∈ A. 3 Example:
1 One die with six sides is rolled. 2 S = {1, 2, 3, 4, 5, 6}. 3 Let A be the event of an even roll 4 Then A = {2, 4, 6}. 5 Let B be the event of a role less than or equal to 3. 6 Then B = {1, 2, 3}. 7 Then 1 ∈ B, 1 /∈ A.
Continued
1 a ∈ A, read a is an element of A.
2 If it is not true that a ∈ A, then we write a /∈ A. 3 Example:
Continued
1 a ∈ A, read a is an element of A. 2 If it is not true that a ∈ A, then we write a /∈ A.
3 Example:
Continued
Continued
1 One die with six sides is rolled.
2 S = {1, 2, 3, 4, 5, 6}. 3 Let A be the event of an even roll 4 Then A = {2, 4, 6}. 5 Let B be the event of a role less than or equal to 3. 6 Then B = {1, 2, 3}. 7 Then 1 ∈ B, 1 /∈ A.
Continued
1 One die with six sides is rolled. 2 S = {1, 2, 3, 4, 5, 6}.
3 Let A be the event of an even roll 4 Then A = {2, 4, 6}. 5 Let B be the event of a role less than or equal to 3. 6 Then B = {1, 2, 3}. 7 Then 1 ∈ B, 1 /∈ A.
Continued
1 One die with six sides is rolled. 2 S = {1, 2, 3, 4, 5, 6}. 3 Let A be the event of an even roll
4 Then A = {2, 4, 6}. 5 Let B be the event of a role less than or equal to 3. 6 Then B = {1, 2, 3}. 7 Then 1 ∈ B, 1 /∈ A.
Continued
1 One die with six sides is rolled. 2 S = {1, 2, 3, 4, 5, 6}. 3 Let A be the event of an even roll 4 Then A = {2, 4, 6}.
5 Let B be the event of a role less than or equal to 3. 6 Then B = {1, 2, 3}. 7 Then 1 ∈ B, 1 /∈ A.
Continued
1 One die with six sides is rolled. 2 S = {1, 2, 3, 4, 5, 6}. 3 Let A be the event of an even roll 4 Then A = {2, 4, 6}. 5 Let B be the event of a role less than or equal to 3.
6 Then B = {1, 2, 3}. 7 Then 1 ∈ B, 1 /∈ A.
Continued
1 One die with six sides is rolled. 2 S = {1, 2, 3, 4, 5, 6}. 3 Let A be the event of an even roll 4 Then A = {2, 4, 6}. 5 Let B be the event of a role less than or equal to 3. 6 Then B = {1, 2, 3}.
7 Then 1 ∈ B, 1 /∈ A.
Continued
Continued
1 Let ⊂ be the relation indicating that every element of the left side set is an element of the right side set.
1 That is, A ⊂ B if, whenever a ∈ A, then a ∈ B. See Fig. 1.
Fig. 1: Illustrating A ⊂ B
B
Continued
1 Let ⊂ be the relation indicating that every element of the left side set is an element of the right side set.
1 That is, A ⊂ B if, whenever a ∈ A, then a ∈ B. See Fig. 1.
Fig. 1: Illustrating A ⊂ B
B
Operations on sets
1 For two sets A and B, let A ∪ B, read “A union B”, be the set of points such that are either in A, or in B, or in both. See Fig. 2.
Fig. 2: Illustrating A ∪ B
S
Continued 1 For two sets A and B, let A ∩ B be the set of points such that are in
both A and B. See Fig. 3.
Fig. 3: Illustrating A ∩ B
S
Dotted lines represent the boundary of A ∩ B
1 read “A and B” or “A intersection B” 2 For points in the intersection, we say that the two events happen
simultaneously.
S
1 read “A and B” or “A intersection B”
2 For points in the intersection, we say that the two events happen simultaneously.
S
Dotted lines represent the boundary of A ∩ B
1 read “A and B” or “A intersection B” 2 For points in the intersection, we say that the two events happen
simultaneously. Introduction:: Statistical Notions: Lecture 01 19 / 1
Continued
1 Continued
1 If the intersection is empty, we say that the two sets are mutually exclusive. See Fig. 4.
Fig. 4: Mutually Exclusive Sets illustrating A ∩ B = ∅
S
Continued
1 Continued 1 If the intersection is empty, we say that the two sets are mutually
exclusive. See Fig. 4.
S
A set’s complement are those points not in the set.
1 Denote the complement of A by A. See Fig. 5.
Fig. 5: Set Complement
A A
1 Note A = A. 2 Note that the middle vowel is “e”.
1 Compliment is something nice said about someone. 2 Not the same thing.
2 Note that the middle vowel is “e”.
A A
A A
1 Compliment is something nice said about someone.
2 Not the same thing.
A A
Example (Hayter) I
Question 1.1.6, page 8: A bag contains balls that are either red or blue and either dull or shiny. What is the sample space when a ball is chosen from the bag?
Solution (Hayter) I
Question 1.1.6, page 8: A bag contains balls that are either red or blue and either dull or shiny. What is the sample space when a ball is chosen from the bag? S = {(red, dull), (red, shiny), (blue, dull), (blue, shiny)}
Introduction:: Set Laws Lecture 01 24 / 1
Distributive Law 1 A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ). See Fig. 6.
Fig. 6: Distributive Law of Sets, showing A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )
S
A
Boundary of B ∩ C is dotted. A ∩ (B ∪ C ) is shaded.
2 By analogy with distributive law for aritheatic 3 Here ∪ is analogous to +, ∩ is analogous to ×. 4 We will see soon that this analogy is apt for probabilities.
S
A
2 By analogy with distributive law for aritheatic
3 Here ∪ is analogous to +, ∩ is analogous to ×. 4 We will see soon that this analogy is apt for probabilities.
S
A
2 By analogy with distributive law for aritheatic 3 Here ∪ is analogous to +, ∩ is analogous to ×.
4 We will see soon that this analogy is apt for probabilities.
S
A
2 By analogy with distributive law for aritheatic 3 Here ∪ is analogous to +, ∩ is analogous to ×. 4 We will see soon that this analogy is apt for probabilities.
Surprising Distributive Law 1 A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C ). See Fig. 7.
Fig. 7: Obscure Distributive Law of Sets A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
S
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Dotted lines bound B ∩ C . Shaded region is A ∩ (B ∪ C ).
2 Here ∩ is analogous to + and ∪ is analogous to ×.
3 Analogy is less compelling, but it is true.
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3 Analogy is less compelling, but it is true.
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3 Analogy is less compelling, but it is true. Introduction:: Set Laws Lecture 01 26 / 1
Proof that A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )
1 To prove that two sets are the same,
1 Take any element of first and show that it is in second. 2 Take any element of second and show that it is in first.
2 Take d ∈ A ∪ (B ∩ C ).
1 Then d ∈ A or d ∈ B ∩ C . 2 If d ∈ A,
1 then d ∈ A ∪ B and d ∈ A ∪ C . 2 Hence d ∈ (A ∪ B) ∩ (A ∪ C).
3 If d ∈ B ∩ C ,
1 then d ∈ A ∪ B and d ∈ A ∪ C . 2 and d ∈ (A ∪ B) ∩ (A ∪ C).
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
1 If d /∈ A, then d ∈ B and d ∈ C , and so d ∈ A ∪ (B ∩ C ). 2 if d ∈ A, then d ∈ A ∪ (B ∩ C ).
1 To prove that two sets are the same, 1 Take any element of first and show that it is in second.
2 Take any element of second and show that it is in first.
2 Take d ∈ A ∪ (B ∩ C ).
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
1 To prove that two sets are the same, 1 Take any element of first and show that it is in second. 2 Take any element of second and show that it is in first.
2 Take d ∈ A ∪ (B ∩ C ).
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
2 Take d ∈ A ∪ (B ∩ C ).
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
2 Take d ∈ A ∪ (B ∩ C ). 1 Then d ∈ A or d ∈ B ∩ C .
2 If d ∈ A,
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
2 Take d ∈ A ∪ (B ∩ C ). 1 Then d ∈ A or d ∈ B ∩ C . 2 If d ∈ A,
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
1 then d ∈ A ∪ B and d ∈ A ∪ C .
2 Hence d ∈ (A ∪ B) ∩ (A ∪ C).
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
3 If d ∈ B ∩ C ,
1 then d ∈ A ∪ B and d ∈ A ∪ C .
2 and d ∈ (A ∪ B) ∩ (A ∪ C).
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C )
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C ) 1 If d /∈ A, then d ∈ B and d ∈ C , and so d ∈ A ∪ (B ∩ C ).
2 if d ∈ A, then d ∈ A ∪ (B ∩ C ).
3 If d ∈ B ∩ C ,
3 Take d ∈ (A ∪ B) ∩ (A ∪ C ) 1 If d /∈ A, then d ∈ B and d ∈ C , and so d ∈ A ∪ (B ∩ C ). 2 if d ∈ A, then d ∈ A ∪ (B ∩ C ).
DeMorgan’s Laws
Fig. 8: Illustrating DeMorgan’s Law (A ∩ B) = A ∪ B
S
Everything outside dotted lines represent (A ∩ B) = A ∪ B
2 Apply the same thing to A and B: (A ∪ B) = A ∩ B.
DeMorgan’s Laws
Fig. 8: Illustrating DeMorgan’s Law (A ∩ B) = A ∪ B
S
Everything outside dotted lines represent (A ∩ B) = A ∪ B
2 Apply the same thing to A and B: (A ∪ B) = A ∩ B.
Example (Hayter) I
Question 1.3.10, page 32: A card is drawn from a pack of cards. A is the event that an Ace is obtained. B is the event that a card from one of two red suites is obtained, and C is the event that a picture card is obtained. What cards to the following events consist of ?
a. A ∩ B
b. A ∪ C
c. B ∩ C
Solution (Hayter) I
Question 1.3.10, page 32: A card is drawn from a pack of cards. A is the event that an Ace is obtained. B is the event that a card from one of two red suites is obtained, and C is the event that a picture card is obtained. What cards to the following events consist of ?
a. A ∩ B A ∩ B = (Ace drawn) ∩ (one of the red suits drawn) = (an ace of hearts OR an ace of diamonds drawn) = {A♥,A♦}. b. A ∪ C A ∪ C = (Jack, Queen, King or Ace of any suit drawn) = {A♥,A♦,A♣,A♠,K♥,K♦,K♣,K♠,Q♥,Q♦,Q♣,Q♠, J♥, J♦, J♣, J♠}.
c. B ∩ C By definition, B ∩ C = (red card drawn) ∩ (card without a picture drawn). Putting these together, B ∩ C = (a red suit drawn that is not a picture card), and, simplifying, B ∩ C = (a red Ace or a red number card drawn) = {A♥, 1♥, . . . , 9♥,A♦, 1♦, . . . , 9♦}.
Solution (Hayter) II
d. A ∪ (B ∩ C ) By definition, A ∪ (B ∩ C ) = (an Ace drawn)∪((a card that is not red drawn)∩( a picture card drawn)). Combining the sets in parentheses, A ∪ (B ∩ C ) = (an Ace drawn) ∪ ( a black picture card drawn). Then A ∪ (B ∩ C ) = ( an Ace or a black picture card drawn) = {A♥,A♦,A♣,A♠,K♠,K♣,Q♠,Q♣, J♠, J♣}.
4 Readings: WMS §2.4, 2.5, 2.6
4 Readings: WMS §2.4, 2.5, 2.6
4 Readings: WMS §2.4, 2.5, 2.6
4 Readings: WMS §2.4, 2.5, 2.6
Section: Probability Theory.
Sample space S
1 A sample space is the set of outcomes (or sample points) from an experiment.
2 Examples:
1 Sequence of multiple coin tosses: get things like s = HTHHTHT ∈ S . 2 A medical experiment tests the effectiveness of various treatments.
1 Get results s = description of the individuals’ health before and after treatment.
3 An obsn on the economy. Get results s = description of all decisions individuals made on how to spend time, money, and other resources.
Probability Theory.: Definitions Lecture 02 34 / 1
Sample space S
2 Examples:
1 Sequence of multiple coin tosses: get things like s = HTHHTHT ∈ S . 2 A medical experiment tests the effectiveness of various treatments.
Sample space S
2 Examples: 1 Sequence of multiple coin tosses: get things like s = HTHHTHT ∈ S .
2 A medical experiment tests the effectiveness of various treatments.
Sample space S
2 Examples: 1 Sequence of multiple coin tosses: get things like s = HTHHTHT ∈ S . 2 A medical experiment tests the effectiveness of various treatments.
Sample space S
Sample space S
A sample space is discrete if it can be put into a list indexed by integers.
1 This property of being able to be put in a list indexed by integers is called countable in more general contexts.
2 Examples:
1 Any finite set is countable. 2 The positive integers are countable: {1, 2, 3, 4, 5, . . .}. 3 The non-negative integers are countable: {0, 1, 2, 3, 4, 5, . . .}. 4 All integers are countable: {0, 1,−1, 2,−2, 3,−3, . . .}.
2 Examples:
1 Any finite set is countable. 2 The positive integers are countable: {1, 2, 3, 4, 5, . . .}. 3 The non-negative integers are countable: {0, 1, 2, 3, 4, 5, . . .}. 4 All integers are countable: {0, 1,−1, 2,−2, 3,−3, . . .}.
2 Examples: 1 Any finite set is countable.
2 The positive integers are countable: {1, 2, 3, 4, 5, . . .}. 3 The non-negative integers are countable: {0, 1, 2, 3, 4, 5, . . .}. 4 All integers are countable: {0, 1,−1, 2,−2, 3,−3, . . .}.
2 Examples: 1 Any finite set is countable. 2 The positive integers are countable: {1, 2, 3, 4, 5, . . .}.
3 The non-negative integers are countable: {0, 1, 2, 3, 4, 5, . . .}. 4 All integers are countable: {0, 1,−1, 2,−2, 3,−3, . . .}.
2 Examples: 1 Any finite set is countable. 2 The positive integers are countable: {1, 2, 3, 4, 5, . . .}. 3 The non-negative integers are countable: {0, 1, 2, 3, 4, 5, . . .}.
4 All integers are countable: {0, 1,−1, 2,−2, 3,−3, . . .}.
2 Examples: 1 Any finite set is countable. 2 The positive integers are countable: {1, 2, 3, 4, 5, . . .}. 3 The non-negative integers are countable: {0, 1, 2, 3, 4, 5, . . .}. 4 All integers are countable: {0, 1,−1, 2,−2, 3,−3, . . .}.
Continued
1 Counterexamples:
1 The set of all sequences of coin flip outcomes is NOT countable.
1 Suppose we have a list of sequences of coin flips: 2 Suppose we have a list of sequences of coin flips:
1: H H H T H T . . . 2: H H T T H T . . . 3: H H H H T T . . . 4: T T H H T H . . . 5: H T T T T T . . . 6: H T H H T T . . .
3 Sequence with diagonal elements reversed (that is, reverse flip j of sequence j) gives a sequence not in the list.
4 TTTTHH . . . in this case. 5 Hence no list can contain all such sequences.
2 Possible real numbers (even on an interval) is NOT countable.
1 Identify H, T with 0, 1 in binomial expansion of numbers in [0, 1) 2 Even after removing sequences leading to idential numbers.
Continued
1 Counterexamples: 1 The set of all sequences of coin flip outcomes is NOT countable.
Continued
1 Suppose we have a list of sequences of coin flips:
2 Suppose we have a list of sequences of coin flips:
Continued
Continued
Continued
4 TTTTHH . . . in this case.
5 Hence no list can contain all such sequences.
Continued
Continued
Continued
1 Identify H, T with 0, 1 in binomial expansion of numbers in [0, 1)
2 Even after removing sequences leading to idential numbers.
Continued
An event is a collection of elements in S . 1 A simple event is one corresponding to a single element of S .
1 Example:
1 Throw one die. 2 Sample space is set of possible outcomes S = {1, . . . , 6}. 3 Event “Get 6” is simple
2 An event is compound if it is not simple.
1 Example: In previous sample space S = {1, . . . , 6}, event “Get an even number” is compound.
3 Example:
1 Throw two die, one red and one green. 2 Sample space is
R1G1 R1G2 R1G3 R1G4 R1G5 R1G6 R2G1 R2G2 R2G3 R2G4 R2G5 R2G6 R3G1 R3G2 R3G3 R3G4 R3G5 R3G6 R4G1 R4G2 R4G3 R4G4 R4G5 R4G6 R5G1 R5G2 R5G3 R5G4 R5G5 R5G6 R6G1 R6G2 R6G3 R6G4 R6G5 R6G6
3 After adding second die, event “Get 6 on red” is no longer simple.
4 A probability measure tells how likely various events are.
1 In symbols the probability of event A is P (A).
1 Example:
1 Throw one die. 2 Sample space is set of possible outcomes S = {1, . . . , 6}. 3 Event “Get 6” is simple
3 Example:
1 Example: 1 Throw one die.
2 Sample space is set of possible outcomes S = {1, . . . , 6}. 3 Event “Get 6” is simple
3 Example:
1 Example: 1 Throw one die. 2 Sample space is set of possible outcomes S = {1, . . . , 6}.
3 Event “Get 6” is simple 2 An event is compound if it is not simple.
3 Example:
1 Example: 1 Throw one die. 2 Sample space is set of possible outcomes S = {1, . . . , 6}. 3 Event “Get 6” is simple
3 Example:
1 Example: 1 Throw one die. 2 Sample space is set of possible outcomes S = {1, . . . , 6}. 3 Event “Get 6” is simple
3 Example:
An event

section: housekeeping subsection: hygene

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