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APMA 3100

Fall 2020 1.1: Set Theory

Chapter 1

Definition: A set is a collection of ____________________. Examples:

𝐴 = {1,2,3,4, . . . } 𝐵 = {2,4,6,8,10} C = the set of all planets in our solar system

Notation:

We use the symbol ___ to show that something is an element of a set.

Examples:

Mercury ____ C. The sun ____ C.

Notation: Definition:

We use 𝐴 ⊂ 𝐵 to say that ___________________________________________. If A is a subset of B, then all the elements in A are also in B. If ______________, then if ____________ , then ____________.

Example:

Using sets A, B, and C above, we could say ___________.

Definition:

The ________ set or ____________ set is the set containing no elements.

Notation:

It is denoted by ______ or _______.

Definition:

The _________________ set or the ________________ is the set of all elements we are considering.

Notation:

It is denoted by ____.

Definition: The ____________ of sets A and B includes all elements in A, B, or both. Notation:

Definition: The _____________________ of sets A and B includes only the elements that are in both A and B. Notation:

Definition: The ______________________ of a set is everything that is NOT in the set. Notation: We denote the complement of set A with either _____ or _____. So if 𝑥 ∈ 𝐴𝑐 then _____________.

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APMA 3100 Example: Illustrate 𝐴 ∪ 𝐵𝑐 .

Fall 2020 Chapter 1 Example: Use set theory notation to express the shaded region:

Definition: A collection of sets is ______________ or _________________ _________________ (ME) if they have no elements in common - that is, if 𝐴𝑖 ∩ 𝐴𝑗 = _____ for 𝑖 ≠ 𝑗 .

Definition: A collection of sets is ______________________ _____________________ (CE) if every element in the universe is contained in at least one of the sets. That is, if 𝐴1 ∪ 𝐴2 ∪ … ∪ 𝐴𝑛 = ____ , then 𝐴1 , 𝐴2 , … 𝐴𝑛 are _____. Example: A and B are _____________________________________________. A and B are also _________________________________________. A, B, and C are ___________________________________________.

Definition: If a collection of sets is both _____ and _____, we call it a __________________.

Example: In the example above, which two sets form a partition?

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APMA 3100

Fall 2020 1.2: Application of Set Theory to Probability SET THEORY Set Element Universal Set

Chapter 1

PROBABILITY

As a set contains elements, an ____________ contains _________________. As a set must contain elements from the universe, an ____________ must contain ________________ from the ________________________________. Example: Roll two dice. What are the possible outcomes? This collection of all the possible outcomes makes up the ________________________________. Definition: An ____________ is a particular collection of outcomes. (just as a set is a particular collection of elements) Example: Which outcomes are included in each of these sets? Event A is rolling a 2 on the first die and a 3 on the second.

Event B is getting a sum of 5.

1.3 Probability Axioms A probability measure 𝑃[∙] is a function that maps events in the sample space to real numbers so that: Axiom 1: For any event A, __________________ Axiom 2: P[S] = ___ Axiom 3: For any countable collection 𝐴1, 𝐴2 , 𝐴3 , . .. of ME events, 𝑃[𝐴1 ∪ 𝐴2 ∪. . . ] = ______________________ Definition: 𝑃[𝐴] is ________________________________________________________________________________________________________________ __________________________________________________________________________________________________ 𝑃[∅] = _____ 𝑃[𝐴𝑐 ] = _____________ 𝑃[𝐴 ∪ 𝐵] = ___________________________ If 𝐴 ⊂ 𝐵, 𝑃[𝐴] ≤ 𝑃[𝐵] When all outcomes have equal probability, 𝑃[𝐴] = 3

APMA 3100 Fall 2020 Example: Roll a single die. What is the probability the die lands on a number less than 3?

Chapter 1

1.4 Conditional Probability Probabilities can change when we get additional information. For example, consider the following questions: Example: Roll two dice. Find the following 𝑃[sum = 4] P[sum = 4|first die lands on 5] 𝑃[sum = 4|first die lands on 1]

The second and third probabilities above are _____________________ __________________________. In these cases, we have eliminated some of the ________________. We have narrowed the ______________________. To calculate the second and third probabilities, we divided the number of outcomes in the ________________________ by the number of outcomes in the ___________________________ Definition: The conditional probability of event A, given event B, is denoted by 𝑃[𝐴|𝐵] and is calculated by 𝑃[𝐴|𝐵] = ________________

We can rewrite the equation above and calculate 𝑃[𝐴 ∩ 𝐵] = _________________________________ Example: A professor gives 2 tests. 75% of the students pass Test 1. 60% pass both tests. 15% fail both. What percentage who pass Test 1 also pass Test 2? Note: This question could be reworded as: Given that ________________________________________, what is the probability they also __________________?

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APMA 3100

Fall 2020

Chapter 1

1.5: The Law of Total Probability and Baye’s Theorem We will introduce this topic with the following example: Suppose it is known that the proportion of people in a community who have a certain disease is .005. A new, cheaper, but less reliable test is now available to diagnose the disease.  If a person has the disease, the probability the test will produce a positive signal is .99.  If a person does not have the disease, the probability that the test will produce a positive signal (“false positive”) is .01. 1. What percentage of people have the disease and test positive? 2. What percentage of people do not have the disease and test positive? 3. What percentage of people test positive? 4. If a person tests positive, what is the probability the person actually has the disease? 5. Interpret your result from #4. Is your answer reasonable? Let Event A be having the disease. Let Event B be testing positive.

In question 3, you discovered and used the Law of Total Probability. The Law of Total Probability tells us that if two sets, A and Ac, form a _________________, and set B intersects those sets (see diagram below), then 𝑃[𝐵] = ________________ + ________________ = __________________________ + _________________________

Note: this can be extended to partitions made of more than two sets. In question 4, you discovered and used Baye’s Theorem. 𝑃[𝐴|𝐵] = ______________ = ___________________________ = _________________________________________ Note this can be extended to more than two sets. 5

APMA 3100

Fall 2020

Chapter 1

1.6: Independence We will again, here, use the fact that 𝑃[𝐴 ∩ 𝐵] = _______________________________. If A and B are independent events – that is, if Event A is not __________________ by Event B, and vice versa, then 𝑃[𝐴|𝐵] = ____________ (because we’re assuming A doesn’t depend on B). This means 𝑃[𝐴 ∩ 𝐵] = ___________ ∙ __________ The reverse is true. If 𝑃[𝐴 ∩ 𝐵] = ___________ ∙ __________ , then A and B are independent. So we say: Definition: Events A and B are independent iff (if and only if) ___________________________________. Example: Flip a fair coin twice. Use the fact that each flip is independent of the other to find the probability of getting heads twice.

Example: The following table shows music preferences of people surveyed from two different regions. Based on the sample, can we conclude that music preference is independent of region? Hint: Let choosing “Music 1” be Event A and being in “Region 1” be Event B.

Region 1 Region 2 Total

Music 1 15 51 66

Music 2 13 25 38

Total 28 76 104

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