Q.4.7 - A First Course in Probability

Question

Suppose that a die is rolled twice. What are the possible values that the following random variables can take on: 

(a) the maximum value to appear in the two rolls; 

(b) the minimum value to appear in the two rolls; 

(c) the sum of the two rolls; 

(d) the value of the first roll minus the value of the second roll? 

Step-by-Step Solution

Answer

1. The maximum value to appear in two rolls are $\{1,2,3,4,5,6\}$
2. The minimum value to appear in the two rolls are $\{1,2,3,4,5,6\}$
3. The sum of the two rolls can be from $2 to 12$
4. The value of the first roll minus the value of the second roll from $-5 to 5$

Step 1 : Given Information (part a)

Let's consider that a die is rolled twice.

The maximum of two rolls is also remains a random variable.

Step 2 : Explanation (part a)

A die includes the numbers of $\{1,2,3,4,5,6\}$. That means there are $36$ combinations can occur.

Since for every $k\in \{1,2,3,4,5,6\}$ ordered pair $(k,k)$ can occur.

Therefore, the maximum value appear in the two rolls is $\{1,2,3,4,5,6\}$

Step 3: Final Answer

The maximum value appear in the two rolls is $\{1,2,3,4,5,6\}$

Step 4 : Given Information (part b)

Let's consider that a die is rolled twice.

The maximum of two rolls is also remains a random variable.

We have to find the minimum value to appear in the two rolls.

Step 5 : Explanation (part b)

Since,$(k,k)$ can occur in two rolls, Where $k\in \{1,2,3,4,5,6\}$.

Therefore, the minimum value to appear in the two rolls are $\{1,2,3,4,5,6\}$

Step 6: Final Answer (Part b)

The minimum value to appear in the two rolls;  $\left\{1,2,3,4,5,6\right\}$

Step 7 : Given Information (part c)

Let's consider that a die is rolled twice.

The maximum of two rolls is also remains a random variable.

We have to find the sum of two rolls.

Step 8 : Explanation (part c)

Both die can take values from $1$to $6$. Therefore the smallest sum is $2$ and the largest sum is $12$.It is trivial to see that every number un between can occur.

The possible values are $\left\{2,3,4,5,6,7,8,9,10,11,12\right\}$

Step 9 : Final Result (part c)

The sum of two rolls are taken from $2$ to $12$

Step 10 : Given Information (part d)

Let's consider that a die is rolled twice.

The maximum of two rolls is also remains a random variable.

We have to find the value of the first roll minus the value of the second roll.

Step 11 : Explanation (part d)

Possible value for first roll $\left\{1,2,3,4,5,6\right\}$

Possible value for second roll$\left\{1,2,3,4,5,6\right\}$

 

The value of first roll minus second roll

 $\left\{-5,-4,-3,-2,-1,0,1,2,3,4,5\right\}$

Step 12 : Final Result (part d)

The value of the first roll minus the value of the second roll from $-5 to 5$