Week 10 - Probability

# Week 10 - Probability
## In Class Exercise
<html>
<div style="float:left">

</div>
<hr color='#EB811B' size=1px width=800px>
</html>
### Danilo Freire
### 5 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>

# In Class Exercise

.font150[
* Four quick questions: 
    - Objective and subjective probability
	- Permutations
	- Combinations
	- Conditional probability
	
* And an interesting example of signalling
]
---

# Question 1: Frequentist x Bayesian Prob.

.font130[
* Identify the following as _frequentist_ (objective) and _Bayesian_ (subjective) probability claims:

- Using her old gradebooks, Professor Lindström determines that the likelihood that one of her students earns an A is 0.18.
    - Professor Turan suspects that today few of his students have arrived to class well prepared.
    - Professor Long learns that over the past five years, 79% of his students have given him “Very Good” or “Excellent” ratings.
    - Professor Lee tells her students that nuclear war between Pakistan and India is unlikely.
    
.orange[5 minutes to answer, 5 minutes for discussion]
]
---

# Question 1: Frequentist x Bayesian Prob.

.font150[
* .orange[Frequentist/objective probability:] the ratio between the number of times the event occurs and the number of trials, in repeated trials under the same conditions

* .orange[Bayesian/subjective probability:] measure of one's subjective belief about the likelihood of an event occurring
]
---

# Answers to Question 1

.font150[
1) Using her old gradebooks, Professor Lindström determines that the likelihood that one of her students earns an A is 0.18.]
--
.font150[
A: .orange[Frequentist.] the professor counts similar events (grades) in repeated trials (courses)
]
--
.font150[
2) Professor Turan suspects that today few of his students have arrived to class well prepared.]
--
.font150[
A: .orange[Bayesian.] Turan assumes this is the case based on one single observation (students' performance today) and his prior beliefs about the students' level of effort 
]
---

# Answers to Question 1

.font150[
3) Professor Long learns that over the past five years, 79% of his students have given him “Very Good” or “Excellent” ratings.]
--
.font150[
A: .orange[Frequentist.] Ratio of comparable events over time
]
--
.font150[
4) Professor Lee tells her students that nuclear war between Pakistan and India is unlikely.]
--
.font150[
A: .orange[Bayesian.] Unique event that cannot be repeated under the same conditions
]
---

# Question 2: Permutations

1) How many ways there are to put 12 people in 5 chairs?

2) And 10 people in 10 chairs?

3) A zip code contains 5 digits. How many different zip codes can be made with the digits 0–9 if no digit is used more than once and the first digit is not 0?

# Answer2 to Question 2

.font150[
1) How many ways there are to put 12 people in 5 chairs?
]
--
.font150[
A: `$\frac{12!}{(12-5)!} = 12 \times 11 \times 10 \times 9 \times 8 = 95,040$`
]
--
.font150[
2) And 10 people in 10 chairs?
]
--
.font150[

```r
factorial(10) # New R command!
```

```
## [1] 3628800
```
]
---

# Answers to Question 2

.font140[
* A zip code contains 5 digits. How many different zip codes can be made with the digits 0–9 if no digit is used more than once and the first digit is not 0? 
]
--
.font140[
A: For the first position, there are 9 possible choices (since 0 is not allowed). After that number is chosen, there are 9 possible choices (since 0 is now allowed). Then, there are 8 possible choices, 7 possible choices and 6 possible choices.

`$$9 \times 9 \times 8 \times 7 \times 6 = 27,216$$`

# Question 3: Combinations

.font150[
* To win a particular lottery game, a player chooses 4 numbers from 1 to 60. Each number can only be chosen once. If all numbers match the 4 winning numbers, regardless of order, the player wins. What is the probability that the winning numbers are 35, 2, 28, and 59?

* .orange[Do not use R to answer this question.]

* .orange[5-10 minutes to write the answer, 5-10 minutes for discussion]
]
---

# Answer to Question 3

.font140[
* We choose 4 numbers from 60. As _we don't care about the order_, we're talking about combinations
]
--
.font140[
* Formula: `$\frac{n!}{(n-k)! k!}$`
]
--
.font140[
* `$\frac{60!}{(60-4)!4!} = \frac{60 \times 59 \times 58 \times 57 \times 56 ... \times 2 \times 1}{56! \times 4! }$`
]
--
.font140[
* We cancel out everything from `$56 \times 55 .... \times 1$` in the nominator and `$56!$` in the denominator
]
--
.font140[
* We're left with `$\frac{60 \times 59 \times 58 \times 57}{4 \times 3 \times 2 \times 1} = 487,635$`

* So the chance of having 35, 2, 28, and 59 as the winning numbers are `$\frac{1}{487635}$`
]
---

# Question 4: Conditional Probabilities

.font140[
* Researchers asked graduate students in Rhode Island about what they thought was the coolest idea in economics: signalling, supply and demand, or opportunity costs. Students of political science, sociology and economics were surveyed. Here are the results:

|                   | Political Science | Sociology | Economics | All |
| :---------------- | :---------------- | :-------- | :-------- | :-- |
| Signalling        | 57                | 87        | 103       | 247 |
| Supply and demand | 50                | 42        | 49        | 141 |
| Opportunity costs | 42                | 22        | 26        | 90  |
| All               | 149               | 151       | 178       | 478 |
]
---

# Question 4: Conditional Probabilities

|                   | Political Science | Sociology | Economics | All |
| :---------------- | :---------------- | :-------- | :-------- | :-- |
| Signalling        | 57                | 87        | 103       | 247 |
| Supply and demand | 50                | 42        | 49        | 141 |
| Opportunity costs | 42                | 22        | 26        | 90  |
| **All**           | 149               | 151       | 178       | 478 |

* What is the probability that someone is a political scientist given that he thinks supply and demand is the coolest idea in economics?

* What is the probabily that someone likes signalling theory given that she studies economics?

]
---

# Answer 4

|                   | Political Science | Sociology | Economics | All |
| :---------------- | :---------------- | :-------- | :-------- | :-- |
| Signalling        | 57                | 87        | 103       | 247 |
| Supply and demand | 50                | 42        | 49        | 141 |
| Opportunity costs | 42                | 22        | 26        | 90  |
| **All**           | 149               | 151       | 178       | 478 |

* What is the probability that someone is a political scientist given that he thinks supply and demand is the coolest idea in economics?
]
--
.font120[
* `$\frac{50}{141} \approx 35\%$`
]
--
.font120[
* What is the probabily that someone likes signalling theory given that she studies economics?
]
--
.font120[
* `$\frac{103}{178} \approx 58\%$`
]
---

# Answer 4 - Signalling

* ... and indeed it is!

* Congratulations if you've answered all questions correctly!

* .orange[But before you leave...]

* Professor Skarbek knows a lot about signalling and he has great stories to tell. Take a look at the picture on the next slide and think what is the signal being sent and to whom. 
]
---

---

# See you next week!