Week 10 - Probability

# Week 10 - Probability
## Axioms, Permutations and Combinations
<html>
<div style="float:left">

</div>
<hr color='#EB811B' size=1px width=800px>
</html>
### Danilo Freire
### 1 April 2019

---

<style>

.remark-slide-number {
  position: inherit;
}

.remark-slide-number .progress-bar-container {
  position: absolute;
  bottom: 0;
  height: 6px;
  display: block;
  left: 0;
  right: 0;
}

.remark-slide-number .progress-bar {
  height: 100%;
  background-color: #EB811B;
}

.orange {
  color: #EB811B;
}
</style>

# Chapter 6 - Probability

* Conditional probability

* Bayes' theorem

* Law of large numbers

* Probability distributions
]
---

# Today's Agenda

* The three probability axioms

* Permutations

* Combinations
]
---

# Why Probability?

.font150[
* .orange[Probability rules our lives]
]
--
.font150[
* It is everywhere
]
--
.font150[
* However, humans are _really bad_ at interpreting probabilities

* Even worse at calculating (estimating) probabilities
]
---

# Why Probability?

# Why Probability?

# Why Probability?

* What are the chances you win the lottery?

* What is the probabilty of getting an A in POLS1600?
]
---

# What Is Probability

* Preview: later we will use probability to make statements about uncertainty in data analysis (chapter 7)
]
---

# Two Concepts of Probability

.font150[
* .orange[Frequentist]: long-run frequency of events
  - Ratio between the number of times the event occurs and the number of trials
  - Example: coin flips
]
--
.font150[
* .orange[Bayesian]: subjective belief about the likelihood of event occurrence
  - Evidence based belief
  - Often more sensible philosophy in political world
]
---

# Thomas Bayes

# The Theory That Would Not Die

# Egon Pearson

# Ronald Fisher

---

# Important Terms

.font150[
* .orange[Experiment]: an action or a set of actions that producestochastic events of interest
  - coin toss, rolling a die, voting in an election
  
* .orange[Sample Space] `$(\Omega)$`: all possible outcomes of the experiment
  - {heads, tails}, {1,2,3,4,5,6}, {abstain, Clinton, Trump}
  
* .orange[Event]: subset of sample space
  - tails, 1, vote for Trump
]
---

# Example

.font150[
What is the experiment, sample space, and one event for coin flips or pulling a single card out of a deck of 52? 
]
---

# Example

.font140[
Probability of event A `$= P(A) = \frac{\text{number of elements in A}}{\text{number of elements in sample space}}$`

Probability of heads `$= P(H) = \frac{heads}{heads + tails} = \frac{1}{2}$`
]
--
.font140[
What is the probability of 3 heads in 3 flips?

Sample space?]
--
.font140[
`$$\Omega = \{HHH,HHT,HTH,THH, HTT, THT, TTH, TTT\}$$`
What is the event space we are interested in?]
--
.font150[
`$$\{HHH\}$$`
]
---

# Example

`$$P(3 H) = \frac{\{HHH\}}{\{HHH,HHT,HTH,THH, HTT, THT, TTH, TTT\}}$$`
`$$P(3 H) = \frac{1}{8}$$`
]
---

# Another Example

Sample space?]
--
.font140[
`$$\Omega = \{HHH,HHT,HTH,THH, HTT, THT, TTH, TTT\}$$`
What is the event space we are interested in?]
--
.font140[
`$$\{HHH, HTH, THH, HHT\}$$`]
--
.font140[
`$$P(2 H) = \frac{4}{8}$$` 
]
---

# Axioms (Rules) of Probability

.font140[
From 3 axioms, the entire probability theory can be built
]
.font140[
* Probability of any event is non-negative:
 `$$P(A) \ge 0$$`
 
* Prob. that one of the outcomes in the sample space occurs is 1
`$$P(\Omega) = 1$$`

* .orange[Addition Rule:] If events A and B are mutually exclusive (_meaning only one can happen at a time_), then
`$$P(A\ \textsf{or}\ B) \ = \ P(A) + P(B)$$`
]
---

# Useful Rules of Probability

* Sometimes it is easier to calculate the probability of an event by using its complement
]
---

# Useful Rules of Probability

.font150[
What is the probability of having at least one tails in 3 coin flips?]
--
.font150[
`$$\Omega = \{HHH,HHT,HTH,THH, HTT, THT, TTH, TTT\}$$`]
--
.font150[
P(at least one T) = `$\frac{7}{8}$`

P(at least one T) = `$1 - P(HHH) = 1 - \frac{1}{8}$`
]
---

# Example

`$$\Omega = ?$$`]
--
.font150[
`$$P(Queen) = \frac{4}{52} = \frac{1}{13}$$`
]
---

# Permutation: When Order Matters

* How many possibilities are there to arrange letters A,B,C?]
--
.font150[
* Three outcomes: A, B, C and three draws]
--
.font150[
* First draw: A, B, or C

* Second draw: two possibilities

* Third draw: one left

* 3 x 2 x 1 possibilities 
]
---

# Permutation

.font150[
* Permutations count many ways we can .orange[order] _k_ objects out of a set of _n_ unique objects

`$$_{n}P_{k} = n \times (n-1) \times (n-2) \times ... \times (n-k + 1) = \frac{n!}{(n-k)!}$$`

* What does _n!_ stand for?]
--
.font150[
* n! = n-factorial = `$n \times (n-1) \times (n-2) \times ... \times (n-n+1)$`

* `$3! = 3 \times 2 \times 1$`

* .orange[Note:] `$0! = 1$`
]
---

# Permutation Example

.font150[
* How many ways can we arrange four cards out of the 13 spades in our card deck?
]
--
.font150[
* first draw = ?
]
--
.font150[
* `$13 \times 12 \times 11 \times 10$`]
--
.font150[
* `$\frac{13!}{(13-4)!} = \frac{13!}{9!} = \frac{13 \times 12 \times 11 \times ... \times 2 \times 1}{9 \times 8 \times ... \times 2 \times 1} =$`

* `$13 \times 12 \times 11 \times 10 = 17,160$`
]
---

# Combinations

* So with respect to combinations of 3 out of 26 letters, ABC, BAC, CAB, etc are the same

* There are .orange[always] fewer combinations than permutations

* Draw 2 out of letters ABC. Permutations:
]
--
.font140[
* AB, AC, BA, BC, CA, CB =  `$\frac{3!}{1!}$`

* Combinations:]
--
.font140[
* AB, AC, BC
]
---

# How to Calculate Combinations

* Get rid of counts of different arrangements of same combination: divide by `$k!$`

* `$_{n}C_{k} = {n \choose k} = \frac{_{n}P_{k}}{k!} = \frac{n!}{k!(n-k)!}$`]
--
.font150[
* Why divide by `$k!$` ?]
--
.font150[
* for two sampled elements, we have `$2!(= 2×1 = 2): A, B = AB, BA$`
]
---

# Combination Example: Winning the Lottery

* Pick five numbers between 1 and 70

* Probability of getting 5 correct?

* Size of the sample space?
]
--
.font120[
`$${70 \choose 5} = \frac{70!}{5! \times (70-5)!} =  \frac{70!}{5! \times 65!}$$`
]
--
.font120[$$12,103,014$$]
---

# How To Solve in R

```r
choose(70, 5) # from 70 numbers, choose 5
```

```
## [1] 12103014
```
]
---

# Questions?

<html><div style='float:left'></div><hr color='#EB811B' size=1px width=720px></html>  
---

# See you in a bit!