Instructor: Colin Grudzien

## Instructions: We will work through the following series of activities as a group. ## Activities: ### Activity 1: Set relationships #### Exercise 1: Suppose we are given the following sets. ```{r} A <- c(1, 2, 3, 4) B <- c(3, 4, 5, 6, 7) ``` Can you compute the following? 1. $A\cap B$ 2. $A\cup B$ 3. $A \setminus B$ What is the probability of randomly selecting an odd number from each of the above sets? #### Exercise 2: Suppose that our total space of outcomes is given by $\Omega = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and we have a random trial in which we select one of these numbers with equal probability. Using our probability relationships, what is the probability of selecting a number in $A$ given that it is a number in $B$? ### Activity 2: basic probability spaces In the following you will need the `prob` package in R: ```{r, message=FALSE} require(prob) ``` The following are standard experiments in the `prob` package that we use with the simple probability model. ``` urnsamples(x, size, replace = FALSE, ordered = FALSE, ...), tosscoin(ncoins, makespace = FALSE), rolldie(ndies, nsides = 6, makespace = FALSE), cards(jokers = FALSE, makespace = FALSE), roulette(european = FALSE, makespace = FALSE). ``` In the following exercises, we will compute some complex conditional probabilities for these examples. #### Exercise 1: Suppose we roll 4 fair dies, compute the probability space as in the above and compute all the probabilities generally. #### Exercise 2: From the above solution, compute the probability that the sum of all dice is greater than 20. #### Exercise 3: Now compute the probability that the total of all dies is greater than 20, given that the first two rolls are greater than 4.