Setup

library(tidyverse)

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First, let’s get immersed in a 19th century method of computing probabilities using a normal curve table.

Watch https://www.youtube.com/watch?v=xI9ZHGOSaCg

It’s only 11 minutes.

The Modern Method

What is the probability that a pizza will be less than 16 inches in diameter if the diameters of pizzas is normal with a mmean of 16.3 and a standard deviation of .2?

Solution

pnorm(16,mean =  16.3, sd = .2)

## [1] 0.0668072

Do this using rdrr.io

The point is that anyone who can connect to the internet has immediate access to this level of R functionality. There is no need to install R. Do this with your phone.

Another Problem

Given the same assumptions as above, find the probability that a pizza has a diameter greater than 16.5?

Solution

1 - pnorm(16.5, mean = 16.3, sd = .2)

## [1] 0.1586553

YAP (Yet Another Problem)

Given the same assumption as above, what is the probability that a pizza has a diameter between 15.95 and 16.63?

pnorm(16.63, mean = 16.3, sd = .2) - pnorm(15.95, mean = 16.3, sd = .2)

## [1] 0.9104694

ChatGPT

I asked ChatGPT to solve the problem. Here is the conversation.

https://sharegpt.com/c/A506DPO.

ChatGPT produced the correct code, but the numerical answer was different.

Related Functions: p, d, q, and r.

Read https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/Normal.

What questions are answered by these functions?

I am at the position .7 on the z-axis. How large is the area under the curve to the left of me?

pnorm(.7)

## [1] 0.7580363

How about to the right of me?

1 - pnorm(1.3)

## [1] 0.09680048

I see that the area under the curve to the left of me is .75. Where am I?

qnorm(.75)

## [1] 0.6744898

How high above me is the curve?

dnorm(.75)

## [1] 0.3011374

I want 1,000 random numbers from the standard normal distribution.

x = rnorm(1000)
summary(x)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -2.939300 -0.644394  0.004504  0.011788  0.702654  2.683616

Other Distributions

The functions p, d, q, and r exist for every theoretical distribution. The “norm” is replaced by something else for another distribution.

For example, in the case of the binomial distribution, we have the following.

• pbinom()
• dbinom()
• qbinom()
• rbinom()