# 1/30/2023

## Complete our discussion of from Friday - Empirical Rule

## Final Remarks on Categorical Data (for now)

- See CategoricalSummaries.R
- Watch the video GoogleDrive/R videos and scripts for Kenyon College/MosaicPlots-final
- One last Example - Simpson's Paradox

## Introduction to Probability

Relative Frequency Interpretation

- The probability of an event A, say P(A), is the long-run relative frequency of the event. That is, we independently repeat a random experiment n times and keep track of the fraction of experiments in which the event A occurs. The relative frequency interpretation of P(A) is then the long-run proportion of independent experiments in which A occurs.

Logical Probability

- If a random experiment can result in any one of n equally likely outcomes, then the probability of an event A, say P(A), is equal to one over the number of equally likely outcomes. i.e., P(A)=1/n.

Subjective Probability

- The probability of an event, say P(A), is a number between 0 and 1, inclusive, which measures an individual’s degree of belief in the event.

## Using Simulation to Estimate Probabilities

- Demonstration with R (flipping an unbalanced coin 1000 times - see ProbSim.R)
- Small Group Activity - Estimating Probabilities
- Tossing and spinning coins
- Rolling dice
- Dealing cards

## Basic Probability Definitions

- The
__sample space S__ of a random experiment is the set of all outcomes.
- A
__random variable__ is a variable that takes on numerical values determined by the outcome of a random experiment.
- An
__event__ is a subset of S.

## Example - Rolling a pair of dice

- Identify the sample space.
- Find the probability that the number of dots on the upward facing sides is 6.

## Probabilities in a Finite Sample Space

- Assign probabilities to each basic outcome.
- Compute the probability of the event of interest by summing the probabilities of the basic outcomes making up the event.

## Addition Rule for Disjoint Events

- Two events, say A and B, are called
__disjoint events__ if they have no outcomes in common. If events A and B are disjoint, then P(A or B) = P(A) + P(B).
- Find the probability of getting a 2 or a 6 when rolling two dice.

## Complement Rule

- If A is any event, the set of basic outcomes that are not in A is called the
__complement of A__ and denoted A^{c}.
- For any event A, the probability that A does not occur is P(A
^{c}) = 1 - P(A)
- Find the probability of getting an even number when rolling a pair of dice.

## Multiplication Rule for Independent Events

- Events A and B are
__independent__ if knowledge of the occurrence of one event does not change the probability that the other occurs.
- If A and B are independent events, then P(A and B) = P(A) x P(B)

## What is the probability of getting two heads when tossing a coin twice?

## What is the probability of getting n heads when tossing a coin n times?

## Please read sections 3.1 and 3.2 for class on Wednesday

## Reminder: Our first quiz will be held on Friday and it will cover the material in Chapters 1 and 2.