ECON 3818

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# ECON 3818
## Chapter 12
### Kyle Butts
### 27 September 2021

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## Chapter 12: Introducing Probability

---
# Randomness

What is randomness? A phenomenon is .hi.purple[random] if:
      
1. Individual outcomes are uncertain

2. Has a distributions of outcomes in a large number of repetitions
      
.it[Example]: A coin toss

---
# Probability

.hi.cranberry[Probability]: proportion of times a particular outcome would occur in a very long series of repetitions. 
      
- .it[Example]: What is the .cranberry[probability] of a coin landing on heads?

For a given observation, the .cranberry[probability] that an .coral[event] occurs is:
  
`$$\frac{\text{Number of ways event could occur}}{\text{Number of total possible outcomes}}$$`

---
# Probability and Randomness

---
# Probability Models

We think of probability utilizing a particular framework, first we define a few useful terms:
      
-  .hi.kelly[Sample Space]: set of all possible outcomes

-  .hi.coral[Event]: outcome (or set of outcomes) of a random phenomenon
    - .coral[Event] is a subset of the .kelly[sample space]

-  .hi.cranberry[Probability Model]: assigns a probability to every .coral[event] in the .kelly[sample space]

---
# Probability: Example

Say we roll two six-sided die, the following would be our .kelly[sample space]:

Each outcome is equally likely, specifically each outcome has .cranberry[probability] of 1/36

---
# Clicker Question

If I roll two six-sided die, what is the .cranberry[probability] I roll a one and a two?

---
# Set Notation

`$\coral{A} = \{1,2,3\}$`, `$\coral{B}=\{3,4,5\}$`, `$\coral{C}=\{1,2,3,4,5,6\}$`, `$\coral{D}=\{4\}$`

`$\in$`: "belongs to"
      
-  Example: `$1 \in \coral{A}$`
      
`$\notin$`: "does not belong to"
      
-  Example: `$4 \notin \coral{A}$`
      
`$\cup$`: Union; combination of two or more sets; "or"
      
-  Example: `$\coral{A} \cup \coral{B} = \{1,2,3,4,5\}$`
      
`$\cap$`: Intersection; overlap of two or more sets; "and"
      
-  Example: `$\coral{A} \cap \coral{B} = \{3\}$`

---
# Set Notation, cont.

`$\coral{A} = \{1,2,3\}$`, `$\coral{B}=\{3,4,5\}$`, `$\coral{C}=\{1,2,3,4,5,6\}$`, `$\coral{D}=\{4\}$`

`$\coral{A}^c$`: "A complement"
      
-  Example: `$\coral{A}^c = \{x: x \notin \coral{A} \} = \{4,5,6\}$`

-  Interpreted as "not .coral[A]"
      
`$\subseteq$`: Subset
      
-  Example: `$\coral{A} \subseteq \coral{C}$`, however `$\coral{C} \not\subseteq \coral{A}$`.
      
`$\coral{\emptyset}$`: is the null or empty set

- contains nothing 
      
`$\coral{A} \cap \coral{D}= \emptyset$`: Disjoint

---
# Clicker Question

Given the following sets, `$\coral{A}=\{5,10,15,20\}$` and `$\coral{B}=\{1,2,3,4,5\}$`
Which of the following is true?

<ol type = "a">
  <li>$\coral{A} \cup \coral{B} = \{1,2,3,4,5,10,15,20\}$</li>
  <li>$\coral{A} \cup \coral{B} = \{5\}$</li>
  <li>$\coral{A} \cap \coral{B} = \{5\}$</li>
  <li>$\coral{A} \cap \coral{B} = \{1,2,3,4,5,10,15,20\}$</li>
  <li>Both a. and c.</li>
</ol>

---
# Axioms of Probability

Let `$\coral{A}$` and `$\coral{B}$` be .coral[events], and `$P(\coral{A})$` and `$P(\coral{B})$` are the .cranberry[probability] of those outcomes. We have a set of rules:

1. Any .cranberry[probability] is a number between 0 and 1

2. All possible outcomes together must have the .cranberry[probability] of 1

3. If two .coral[events] are disjoint,

`$$P(\coral{A} \cap \coral{B})=0 \implies P(\coral{A} \cup \coral{B}) = P(\coral{A}) + P(\coral{B})$$`

4. `$P(\coral{A}^c)=1-P(\coral{A})$`

---
# Clicker Question
Given the three following scenarios:

- A person is randomly selected. .coral[A] is the .coral[event] they are under 18. .coral[B] is the .coral[event] they are over 18.
- A person is selected at random. .coral[A] is the .coral[event] that they earn more than $100,000 per year. .coral[B] is the .coral[event] that they earn more than $250,000.
- A pair of dice are tossed. .coral[A] is the .coral[event] that one of the die is a 3. .coral[B] is the .coral[event] that the sum of two dice is 3.

In which cases are the .coral[events], A and B, disjoint?

---
# De Morgan's Law

.hi.slate[De Morgan's law] of union and intersection. For any two finite sets .coral[A] and .coral[B]:
      
1. `$(\coral{A} \cup \coral{B})^c = \coral{A}^c \cap \coral{B}^c$`

2. `$(\coral{A} \cap \coral{B})^c = \coral{A}^c \cup \coral{B}^c$`

---
# De Morgan's Law Example

Let `$\kelly{S}=\{j, k, l, m, n\}$` and `$\coral{A}=\{j, k, m\}$` and `$\coral{B}=\{k, m, n\}$`

1. `$(\coral{A} \cup \coral{B})^c = (\coral{A}^c \cap \coral{B}^c)$`

<ol type="i">
    <li>
      $(\coral{A} \cup \coral{B})= \{j, k, m, n\} \implies (\coral{A} \cup \coral{B})^c=\{l\}$
    </li>
    <br/>
    <li>
      $\coral{A}^c = \{l, n\}$ and $\coral{B}^c=\{j, l\} \implies \coral{A}^c \cap \coral{B}^c = \{l\}$
    </li>
  </ol>

2. `$(\coral{A} \cap \coral{B})^c = \coral{A}^c \cup \coral{B}^c$`
  <ol type="i">
    <li>
      $ (\coral{A} \cap \coral{B})=\{k,m\} \implies (\coral{A} \cap \coral{B})^c=\{j, l, n\} $
    </li>
    <br/>
    <li>
      $ \coral{A}^c \cup \coral{B}^c=\{l, n\} \cup \{j, l\} \implies \coral{A}^c \cup \coral{B}^c = \{j, l, n\} $
    </li>
  </ol>

---
# Random Variables

.hi.purple[Random variable]: variable whose value is a numerical outcome of a random phenomenon
      
- .hi.purple[Random variables] can be .hi[discrete] or .hi[continuous]
      
.it[Example:] Coin toss
      
- `$X$` can be defined as the number of heads we see in two tosses:
  
  - `$X$` is a discrete random variable; `$X=0,1,2$`

.hi.cranberry[Probability distribution]: tell us what values random variable X can take, and how to assign probabilities to those values

---
# Example

Flip a coin two times

.kelly[Sample space]:
      
- {.coral[(Head, Tail)], .coral[(Head, Head)], .coral[(Tail, Head)], .coral[(Tail, Tail)]}
      
What is the .cranberry[probability] of each .coral[event]?

---
# Clicker Question

If I toss a coin two times, and `$X$` is the number of heads, then what is `$P(\coral{X=2})$`?

---
# Additional Examples

Still flipping a coin twice, what is the .cranberry[probability] of getting at least one head?
      
`$P(\coral{X=1}) + P(\coral{X=2}) = 1-P(\coral{X=0})$`

- Now I only have to calculate one .cranberry[probability]!

---
# Additional Dice Example

What is the .cranberry[probability] of rolling a 7, 11, or double when rolling two dice?

- Axiom 3 tells us we can find probabilities simply by adding if the .coral[event] is disjoint.

.coral[Roll a 7]: `$\left\{(1,6), (6,1), (2, 5), (5,2), (3,4), (4,3) \right\}$`

.coral[Roll a 11]: `$\left\{(5,6), (6,5) \right\}$`

.coral[Roll a double]: `$\left\{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6) \right\}$`

- Are all 3 .coral[events] disjoint?

---
# Additional Dice Example

.coral[Roll a 7]: `$\left\{(1,6), (6,1), (2, 5), (5,2), (3,4), (4,3) \right\}$`

.coral[Roll a 11]: `$\left\{(5,6), (6,5) \right\}$`

.coral[Roll a double]: `$\left\{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6) \right\}$`

`$P(\coral{7}) + P(\coral{11}) + P(\coral{\text{doubles}})=\frac{6}{36}+\frac{2}{36}+\frac{6}{36} \approx 0.4$`