In jrminter/statshelpR: Convenience functions for statistical analysis

knitr::opts_chunk$set(echo = TRUE, warning=FALSE)
suppressWarnings(suppressMessages(suppressPackageStartupMessages(library(ggplot2))))

Introduction to probability

This example is adapted from an example in the first 9:30 min from the OpenIntro YouTube video.

Start with a fair die. It has six sides. What is the probability of rolling a 1?
$P(1) = \frac{1}{6}$. Indeed the probability of all the other numbers is $P(n) = \frac{1}{6}$.

We can also ask, what is the probability of rolling a 1 or a 2? $P(1 \ or\ 2) = \frac{2}{6} = \frac{1}{3}$ since all of the numbers are equally probable.

Probability is used to answer the question of how likely something is to happen. Usually it is used to model random processes. It can also be used for non-random processes. The stock market is one such example. It looks random but each event is the result of someone's choice about what they want to buy or sell. But the system as a whole looks random.

Disjoint or mutually-exclusive event

First we consider disjoint or mutually-exclusive events. Rolling a single die is one example. We only get 1 result for a single roll. There is a rule associated with disjoint events, like rolling a die. If we want to know:

$P(1\ or\ 2) = P(1) + P(2) = \frac{1}{3}$

because each of the outcomes has the same probability. We could also extend this for more mutually exclusive outcomes such as

$P(1\ or\ 2\ or\ 3\ or\ 5) = P(1) + P(2) + P(3) + P(5) = \frac{4}{6} = \frac{2}{3}$.

We are using what is called the Addition Rule for Disjoint Events.

Next we want to know what happens with events that are not Disjoint.

Dealing with events that are not disjoint

Using a standard card deck

We want to consider an example where we draw cards from a standard card deck, shown below.

library(png)
library(grid)
img <- readPNG("./inc/std-card-deck.png")
grid.raster(img)

Problem Statement

Compute the probability of drawing a diamond or a face card

A Venn Diagram helps

img <- readPNG("./inc/card-example-venn-diagram.png")
grid.raster(img)

Note the following probabilities:

$P(diamond) = P(D) = \frac{13}{54} = \frac{1}{4} = 0.25$
$P(face\ card) = P(FC) = \frac{12}{52} = \frac{3}{13} \approx 0.231$

If we want to know $P(diamond\ or\ face\ card)$ we can't use the additivity rule for disjoint events because not all face cards are diamonds.

The General Additivity Rule

$P(A \ or\ B) = P(A) + P(B) - P(A\ and \ B)$

so our answer is

$P(D \ or\ FC) = \frac{13}{52} + \frac{12}{52} - \frac{3}{52} = \frac{22}{52} = \frac{11}{26} \approx 0.423$

and is consistent with our Venn Diagram above.

jrminter/statshelpR documentation built on May 2, 2020, 12:08 a.m.