Day 11

We finally made it to statistics y’all

Adressing Changes

Moving forward, I’ll be providing more regular practice problems beyond your homework and QOTD
- The segments that we’re moving into aren’t generally intuitive, don’t trick yourself
Statistical theory is best shown simulations and pictures
- I’ll be leaning on the blackboard more
Some of these concepts will get fairly abstract
- Don’t hesitate to ask questions BUT
- We actually have to limit questions more as we progress
- Attend office hours, email me, go to the help lab

Article on Probabilistic Thinking

Basic Concepts of Probability I

Why do we study probability in statistics?

Probability: A number between \(0\) and \(1\) that tells us how likely a given “event” is to occur

Probability equal to \(0\) means the event cannot occur

\[P(x)=0\]

Probability equal to \(1\) means the event must occur

\[P(x)=1\]

Probability equal to \(1/2\) means the event is as likely to occur as it is to not occur

\[P(x)=0.5\]

Probability close to 0 (but not equal to 0) means the event is very unlikely to occur

The event may still occur, but we’d tend to be surprised if it did
- \(x=\{\text{A shark attack on a beach in the U.S.}\}\)
  - \(P(x) \approx 0.00000008\)
  - \(P(x) \ne 0\)

Probability close to 1 (but not equal to 1) means the event is very likely to occur

The event may not occur, but we’d tend to be surprised if it didn’t
- \(x=\{ \text{You lose the national lottery you bought a ticket for} \}\)
  - \(P(x) \approx 0.999999997\)
  - \(P(x) \ne 1\)
    - Do not gamble

Probability Terminology

To study probability formally, we need some basic terminology

Experiment (in context of probability):

An activity that results in a definite outcome where the observed outcome is determined by chance

Sample space:

The set of ALL possible outcomes of an experiment; denoted by \(S\)

Flip a coin once

\[S = \{H, T\}\]

Randomly select a person and then determine blood type

\[S = \{A, B, AB, O\}\]

Event

A subset of outcomes belonging to sample space \(S\)
A capital letter towards the beginning of the alphabet is used to denote an event
- i.e. \(A\), \(B\), \(C\), etc.

Suppose we flip a coin twice

\[S = \{HH, HT, TH, TT\}\]

Let \(A\) be the event we observe at least one tails

\[A = \{HT, TH, TT\}\]

Let \(B\) be the event we observe at most one tails

\[B = \{HH, HT, TH\}\]

Simple event: An event containing a single outcome in the sample space \(S\)

\[S = \{HH, HT, TH, TT\}\]

\(A = \text{we observe two heads} = \{HH\}\)

Simple event

Compound event: An event formed by combining two or more events (thereby containing two or more outcomes in the sample space \(S\))

\[S = \{HH, HT, TH, TT\}\]

\(B = \text{we observe a head in the first or in the second flip} = \{HT, TH, HH\}\)

Compound event

Probability Methods

There are 3 general views of probability

Consider these as methods of assigning probabilities to events

Subjective Probability

Probability is assigned based on judgement or experience

i.e. expert opinion, personal experience, “vibe math”

A doctor assessing the chance of a patient recovering from a medical procedure
A managerial team estimating the probability a project will achieve technical success

This probability may not be expressed in an actual number; instead, we may say “low”, “high”, “almost certain”, etc.

Classical Probability

Make some assumptions in order to build a mathematical model from which we can derive probabilities

It’s not vibe math but it can definitely feel like it

Suppose we want to put a probability on the event of observing “tails” in one flip of a coin. We might assume the following:

2 possible outcomes: “heads” or “tails”
The coin is “fair”
- i.e. heads & tails have an equal chance of occurring

Based on our model:

\[\text{the probability of observing tails} = P(\text{tails}) = {1\over 2}\]

Note: This is an example of an equally-likely probability model (i.e., all possible outcomes are equally likely to occur) where:

\[P(A) = \frac{\text{number of outcomes in event } A}{\text{total number of outcomes in } S}.\]

Here \(P(A)\) denotes the probability of the event \(A\)

Relative or Empirical Probability

Think of the probability of an event as the proportion of times that the event occurs

Flipping a tack:

\[P(\text{point up}) = \text{the proportion of all possible flips of the tack where it lands "point up"}\]

We could flip our tack a large number of times (let’s say \(1000\)) and count the number of times it lands point up
This is like a simple random sample (SRS) from the population of all tack flips

\[P(\text{point up}) \approx \frac{\text{number of times "point up" is observed}}{1000}\]

Law of Large Numbers

As the size of our sample (i.e., number of experiments) gets larger and larger:

The relative frequency of the event of our interest gets closer and closer to the true probability
What does this mean to us?
- You can get as close as needed to the true probability by taking a large enough sample
- It is also used to justify the relative frequency view of probability
- This is how some statisticians evaluate new statistical procedures; they simulate many datasets and observe the proportion of time the procedure “works”

Assume that a fair die is rolled (i.e., all outcomes are equally-likely)

What is the sample space?

What’s the probability of rolling a \(5\)?

What’s the probability of rolling an even number?

What’s the probability of rolling a number less than \(3\)?

An automobile insurance company divides customers into three categories: good risks, medium risks, and poor risks. Assume that of a total of \(11,217\) customers, \(7792\) are good risks, \(2478\) are medium risks, and \(947\) are poor risks. As part of an audit, one customer is chosen at random.