Question

Role-playing games Computer games in which the players take the roles of characters are very popular. They go back to earlier tabletop games such as Dungeons & Dragons. These games use many different types of dice. A four-sided die has faces with $1,2,3$and $4$ spots.

(a) List the sample space for rolling the die twice (spots showing on first and second rolls).

(b) What is the assignment of probabilities to outcomes in this sample space? Assume that the die is perfectly balanced.

Short answer

Part (a) After that, the sample sequence contains all potential outcomes on the first and second rolls $1,2,3,4$

Part (b) The probability =$1/16$

Step-by-step solution

Part (a) Step 1. Given Information    

The number of sides in the die equals the number of sides in the die =$4$

The four-sided die's faces = $\left\{1,2,3,4\right\}$

Part (a) Step 2. Concept Used  

A chance process' sample space $S$ is the collection of all potential outcomes. A probability model is a two-part description of a random process that includes a sample space $S$ and probabilities for each result.

Part (a) Step 3. Calculation 

The possible ordered outcomes are listed as an element in a sample space, which is commonly indicated using set notation. It is possible to roll a $1,2,3,4$ and on the first roll. It is possible to roll a $1,2,3,4$on the second roll. After that, the sample sequence contains all potential results on the first and second rolls $\left\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)\right\}$

Part (b) Step 3. Calculation 

The sample space of a sequence of die flips is all possible sequences of outcomes $2^4=16$That is, The sample of a die flip is . Since the given sample space has $16$outcomes$\left\{1,2,3,4\right\}$ Therefore, the probability of all the events mentioned in the given sample space is the same.

Hence, The probability =$\frac{1}{16}$