# 4 part problem about a 4-sided die.

#### Angelaa

##### New member
Alright, so I guess I'm confused as to what the problem is asking me to do.

This is the problem. I'll write exactly as it's shown.

Suppose a 4-sided die is rolled two times and we keep track of the product when the result from the first two die is multiplied by the result from the second die.

A) Draw a four by four grid that demonstrates the different results for the product of the two dice.

B) What is the sample space for the possible products of the two dice?

C) How many different outcomes are possible for the product of the two dice?

D) What outcome occurs most often?

I started out fine. I drew the first four by four grid, but after that I was completely lost. What do they mean when they're asking for the different results for the product of the two dice? What numbers am I supposed to put in the second grid? The rest of the questions are hard for me to answer, too.

#### MarkFL

Staff member
I'm guessing the problem should actually state:

Suppose a 4-sided die is rolled two times and we keep track of the product when the result from the first die is multiplied by the result from the second die.

A) Let the column number represent the result for the first die, and the row number represent the result from the second die, and then each entry in the grid or array will then be the product of the row and column numbers. For example, the upper left entry would be the product $1\,\times\,1=1$ while the lower right would be $4\,\times\,4=16$. Can you fill in the rest?

#### Jameson

Staff member
Hi Angelaa,

Welcome to MHB!

Not trying to highjack the thread, but just want to provide a visual based on MarkFL's explanation. If you are a more visual person this might help you see how you can use his suggestion to help you understand the problem.

 Die 1 rolls 1 Die 1 rolls 2 Die 1 rolls 3 Die 1 rolls 4 Die 2 rolls 1 $$\displaystyle 1 \times 1 = 1$$ Die 2 rolls 2 Die 2 rolls 3 Die 2 rolls 4 $$\displaystyle 4 \times 4 = 16$$

#### Angelaa

##### New member
I'm guessing the problem should actually state:

Suppose a 4-sided die is rolled two times and we keep track of the product when the result from the first die is multiplied by the result from the second die.

A) Let the column number represent the result for the first die, and the row number represent the result from the second die, and then each entry in the grid or array will then be the product of the row and column numbers. For example, the upper left entry would be the product $1\,\times\,1=1$ while the lower right would be $4\,\times\,4=16$. Can you fill in the rest?
So the first part of the problem would look like this?

#### MarkFL

Staff member
Hi Angelaa,

Welcome to MHB!

Not trying to highjack the thread, but just want to provide a visual based on MarkFL's explanation. If you are a more visual person this might help you see how you can use his suggestion to help you understand the problem.

 Die 1 rolls 1 Die 1 rolls 2 Die 1 rolls 3 Die 1 rolls 4 Die 2 rolls 1 $$\displaystyle 1 \times 1 = 1$$ Die 2 rolls 2 Die 2 rolls 3 Die 2 rolls 4 $$\displaystyle 4 \times 4 = 16$$

I started to say "think of the multiplication table you learned in elementary school" but that's a much better way to demonstrate it.

#### MarkFL

Staff member
So the first part of the problem would look like this?

Your grid on the left is what I believe is you are to compose. What does the other grid represent?

edit: Never mind, I see this is the sum, not the product of the two dice.

#### Angelaa

##### New member
I started to say "think of the multiplication table you learned in elementary school" but that's a much better way to demonstrate it.
Thank you I really appreciate it!

Another couple questions though: so for the sample space question, would I answer it like this?

s = {2, 3, 4, 5, 6, 7, 8} and s = {1, 2, 3, 4, 5, 6, 7, 8, 9 , 10, 11, 12, 13, 14, 15, 16}

and for the outcome question, would the answer be 16?

I'm really sorry. I just began my class today and I'm a little slow when it comes to math so I'm sure this must seem really easy and extremely basic to you!

#### MarkFL

Staff member
The sample space is the set of all possible outcomes.

Now, you are only being asked about the product so just use the grid you drew on the left. What elements are present, traditionally listed in ascending order, without repeating any elements?

Yes, the total number of outcomes is $4\,\times\,4=16$, but some of them are repeated. You are being asked for the number of distinct elements in the sample space.

And there is no need to apologize for just having began your statistics course. Every single one of us had our first day too. You are doing well...you are showing your work and making a genuine effort to understand the problem.

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#### Angelaa

##### New member
The sample space is the set of all possible outcomes.

Now, you are only being asked about the product so just use the grid you drew on the left. What elements are present, traditionally listed in ascending order, without repeating any elements?

Yes, the total number of outcomes is $4\,\times\,4=16$, but some of them are repeated. You are being asked for the number of distinct elements in the sample space.

And there is no need to apologize for just having began your statistics course. Every single one of us had our first day too. You are doing well...you are showing your work and making a genuine effort to understand the problem.

Hmm...so... s = {8 x 2} ? I know that might not be right. Am I at least on the right track?

And thank you! I really appreciate your words and help.

#### MarkFL

Staff member
The sample space for the possible products would be the set of unique products that are possible. Referring to your grid, can you list these products in ascending order as a set?

#### Angelaa

##### New member
The sample space for the possible products would be the set of unique products that are possible. Referring to your grid, can you list these products in ascending order as a set?
Oh! I think I see what you're saying. So s = {1,4,9,16}??

#### Jameson

Staff member
Oh! I think I see what you're saying. So s = {1,4,9,16}??
Those are some of the possible products but there are some more. You filled out the table correctly first of all. So now you should add the missing terms into your sample space. 1,4,9 and 16 are all in there but what about 3, 6 or 12? Any others? Basically you should list all of the values you filled into the table you drew, but just don't list anything more than once.

What do you get when you try that?

#### Angelaa

##### New member
Those are some of the possible products but there are some more. You filled out the table correctly first of all. So now you should add the missing terms into your sample space. 1,4,9 and 16 are all in there but what about 3, 6 or 12? Any others? Basically you should list all of the values you filled into the table you drew, but just don't list anything more than once.

What do you get when you try that?

Ohh, see I thought he meant that you just list out the unique numbers- the ones that aren't seen more than once in the table.

So it would be like: s = {1,2,3,4,6,8, 9, 12, 16}, right??

I hope that's right, lol. Thank you for being patient with me as I try to figure this out!

#### MarkFL

Staff member
Yes, that's correct. I do apologize for the ambiguity in my choice of words. I did mean to list each outcome with no repeating of the elements that occur more than once.

So, how many elements are in the sample space (called the cardinality of the set)? This is what part c) is asking.

#### Angelaa

##### New member
Yes, that's correct. I do apologize for the ambiguity in my choice of words. I did mean to list each outcome with no repeating of the elements that occur more than once.

So, how many elements are in the sample space (called the cardinality of the set)? This is what part c) is asking.
No, do not be sorry! You've been a superb help and I really appreciate it!! When it comes to math, I don't always understand things right away so that's probably why I didn't quite see what you were saying.

That answer would be 16, right? And for the outcome question, would it be 5 or is the first table on the right completely irrelevant? If it is irrelevant, I'm assuming the most frequently occurring outcomes are 2, 3, 4, 6, 8, and 12- although the question didn't ask for outcomes, just outcome...so that assumption could be entirely wrong, lol.

#### MarkFL

Staff member
I can completely understand how what I said could be interpreted the way you did. It was a poor choice of words on my part.

In the sample space you correctly gave:

S = (1, 2, 3, 4, 6, 8, 9, 12, 16}

How many elements are in this set?

How many times does each element show up in your grid?

And yes, the second grid is irrelevant, unless you are being asked also about the sums of the two dice.

#### Angelaa

##### New member
I can completely understand how what I said could be interpreted the way you did. It was a poor choice of words on my part.

In the sample space you correctly gave:

S = (1, 2, 3, 4, 6, 8, 9, 12, 16}

How many elements are in this set?

How many times does each element show up in your grid?

And yes, the second grid is irrelevant, unless you are being asked also about the sums of the two dice.
Ah, so 9!

And for most of them, twice. 2, 3, 4, 6, 8, and 12 show up twice in the grid. The other numbers only show up once. Are those the numbers the answer?

#### MarkFL

Staff member
Yes, 9 is the cardinality of the sample space!

Look at your grid again...you'll see one of the outcomes occurs 3 times. Can you find which one?

#### Angelaa

##### New member
Yes, 9 is the cardinality of the sample space!

Look at your grid again...you'll see one of the outcomes occurs 3 times. Can you find which one?
Oh my goodness, I feel so stupid! 4!

I'm so sorry I missed that!!!! Thank you for all of your help and for being patient with me! You have no idea how much I appreciate your help.