# Probability functions intermediate

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In this set of exercises, we are going to explore some of the probability functions in R by using practical applications. Basic probability knowledge is required. In case you are not familiarized with the function `apply`

, check the R documentation.

Note: We are going to use random numbers functions and random processes functions in R such as `runif`

. A problem with these functions is that every time you run them, you will obtain a different value. To make your results reproducible you can specify the value of the seed using `set.seed(‘any number’)`

before calling a random function. (If you are not familiar with seeds, think of them as the tracking number of your random number process.) For this set of exercises, we will use `set.seed(1).`

Don’t forget to specify it before every exercise that includes random numbers.

Answers to the exercises are available here. If you obtained a different (correct) answer than those listed on the solutions page, please feel free to post your answer as a comment on that page.

**Exercise 1**

**Generating dice rolls** Set your seed to 1 and generate 30 random numbers using `runif`

. Save it in an object called `random_numbers`

. Then use the `ceiling`

function to round the values. These values represent rolling dice values.

**Exercise 2**

Simulate one dice roll using the function `rmultinom`

. Make sure `n = 1`

is inside the function, and save it in an object called `die_result`

. The matrix `die_result`

is a collection of 1 one and 5 zeros, with the one indicating which value was obtained during the process. Use the function `which`

to create an output that shows only the value obtained after the dice is rolled.

**Exercise 3**

Using `rmultinom`

, simulate 30 dice rolls. Save it in a variable called `dice_result`

and use `apply`

to transform the matrix into a vector with the result of each dice.

**Exercise 4**

Some gambling games use 2 dice, and after being rolled they sum their value. Simulate throwing 2 dice 30 times and record the sum of the values of each pair. Use `rmultinom`

to simulate throwing 2 dice 30 times. Use the function `apply`

to record the sum of the values of each experiment.

**Exercise 5**

Simulate normal distribution values. Imagine a population in which the average height is 1.70 m with a standard deviation of 0.1. Using `rnorm`

, simulate the height of 100 people and save it in an object called `heights`

.

To get an idea of the values of heights, use the function `summary`

.

**Exercise 6**

90% of the population is smaller than ____________?

**Exercise 7**

Which percentage of the population is bigger than 1.60 m?

**Exercise 8**

Run the following line code before this exercise. This will load a library required for the exercise.

`if (!'MASS' %in% installed.packages()) install.packages('MASS')`

library(MASS)

Simulate 1000 people with height and weight using the function `mvrnorm`

with ` mu = c(1.70, 60) `

and

` Sigma = matrix(c(.1,3.1,3.1,100), nrow = 2) `

**Exercise 9**

How many people from the simulated population are taller than 1.70 m and heavier than 60 kg?

**Exercise 10**

How many people from the simulated population are taller than 1.75 m and lighter than 60 kg?

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