---
tags: introduction to probability, statistics
---

# Expected Values

([home](https://github.com/alexhkurz/introduction-to-probability/blob/master/README.md) ... [next](https://hackmd.io/@alexhkurz/BkW-1p7aL))

(draft)

## Rolling two dice

**Activity:** Roll the two dice 30 times and make a list of the outcomes. Which outcomes are most frequent? Draw a bar chart with columns for the possible outcomes from 2 to 12 and the height of the column indicating how often an outcome happened.

This raises some interesting questions. For example:
- Why appear some outcomes more often than others?
- What would you expect from counting the possibilities?

## Expected values of two dice

**Activity:** Make another bar chart with columns for the possible outcomes from 2 to 12 and the height of the column indicating how many possibilities there are to achieve that outcome. We call this the bar chart of **expected values**.

More questions:
- Should the two bar charts be the same? Or similar? Why are they so different?
- What happens if we do the experiment again and do another 30 rolls of two dice?

## A program to roll dice

This is getting a bit boring to do by hand, so I wrote a little program called [two-dice-1.R](https://github.com/alexhkurz/introduction-to-probability/blob/master/src/two-dice.R). You need to install R to run the program.

number_of_rolls <- 30

die1_sample <- as_tibble(rdunif(number_of_rolls,1,6)) # all values from 1 to 6 are equally likely
die2_sample <- as_tibble(rdunif(number_of_rolls,1,6))
two_dice <- die1_sample + die2_sample # add the values of the two dice

two_dice %>%
ggplot() + geom_bar(aes(x=value)) + scale_x_continuous(name = "two dice", breaks = 2:12)

**Activity:** Run the program. Read the program line by line and try to find out what each line means. A good way to find out what each line in the program means is to change the lines, to execute the program again and to see what happens. Can you make a die that has more than 6 sides?

## Rolling dice large numbers of times

Now we understand the program a bit better, we can use it to roll the two dice as often as we wish at the klick of a button.[^largenumbers]

**Activity:** Run the program for different values of `number_of_rolls`. How large do you have to make `number_of_rolls` so that 7 is reliably the most frequent outcome? How large do you have to make `number_of_rolls` so that the bar chart produced by the program looks similar to the bar chart of expected values?

## Summary

There are many possible questions here. Make a list with questions.

- ...
- ...
- ...

[^largenumbers]: "As often as you wish"? Maybe not quite. Find a value for `number_of_rolls` that is large enough to break `R` (or your patience).