What is the probability that the sum of two dice will yield

[piccolino]

"What is the probability that the sum of two dice will yield a 4 or a 6?" (Word translations/ Probability/ Question 1 page 9)

Let's take the sum 4. Imagine you roll 2 dice (not explicitly said sequentially, nor explicitly said simultaneously; but clearly not the same dice twice).

Imagine you look at them from above the table and make their sum. Can you explain how is the formation showing faces 1 and 3 (adding to 4) different from the formation showing 3 and 1 (also adding to 4)?

MORE IMPORTANTLY IF 1+3 IS DIFFERENT FROM 3+1, SHOULDN'T 2+2 BE ALSO DIFFERENT FROM 2+2, as each number expresses the face of different dice (faces which are only coincidentally identical but still different in the case of 2+2)?

That is, can you explain what is the criterion you use in this question to differentiate dice and to make the order significant. Why shouldn't these dice (if both fair) be indistinguishable for the purpose of sum?

How do you make the difference? Is it by tilting your head at different angles and seeing one dice first and the other second; and then tilting your head again and seeing the dice in reverse order?

Surely I can see that if you throw two dice sequentially, the time-ordered sequences are different, hence the same sum can be obtained by two different throws. But in the context of questions 1 - 5 page 93, I just can't see what criterion was used to make the difference between the rolls, to make order significant, when taking note of the sum of faces.

Thanks!

[pellucide]

"What is the probability that the sum of two dice will yield a 4 or a 6?" (Word translations/ Probability/ Question 1 page 9)

Let's take the sum 4. Imagine you roll 2 dice (not explicitly said sequentially, nor explicitly said simultaneously; but clearly not the same dice twice).

Imagine you look at them from above the table and make their sum. Can you explain how is the formation showing faces 1 and 3 (adding to 4) different from the formation showing 3 and 1 (also adding to 4)?

MORE IMPORTANTLY IF 1+3 IS DIFFERENT FROM 3+1, SHOULDN'T 2+2 BE ALSO DIFFERENT FROM 2+2, as each number expresses the face of different dice (faces which are only coincidentally identical but still different in the case of 2+2)?

That is, can you explain what is the criterion you use in this question to differentiate dice and to make the order significant. Why shouldn't these dice (if both fair) be indistinguishable for the purpose of sum?

How do you make the difference? Is it by tilting your head at different angles and seeing one dice first and the other second; and then tilting your head again and seeing the dice in reverse order?

Surely I can see that if you throw two dice sequentially, the time-ordered sequences are different, hence the same sum can be obtained by two different throws. But in the context of questions 1 - 5 page 93, I just can't see what criterion was used to make the difference between the rolls, to make order significant, when taking note of the sum of faces.

Thanks!

Already discussed here. post47445.html#p47445

[jnelson0612]

Thank you pellucide.

[piccolino]

Thank you, pellucide.

I checked the recommended thread. I don't want to be a pest, but I am not convinced at all.
The other thread explains this in 2 ways:

1- assuming that there is a difference bt. the dice (e.g one red one white) when none is stated; this assumption seems to me the exact think that is otherwise always recommended: not to make any assumptions; and then, if you make the assumption that one is red and one white, why not also make the assumption that the other is red, and one is white and create the reverse case.

2- assuming that the dice are rolled in order, which may be true but also not explicitly stated (hence again, assumed) and still, irrelevant for the sum.

Anyway, if one dice is red and one white, or one first and one second - the question of why is 2+2 not different than 2+2 is still unanswered.

I don't want to create a controversy here but to clarify this for ever an put it to rest forever. In the grand context of gmat, it's a petty question but I would not like to meet with it on the exam.

Since we all seem to submit to ETS dogma .. did you ever see this question and explanation on an ETS provided material? Thank you!

[pellucide]

1- assuming that there is a difference bt. the dice (e.g one red one white) when none is stated; this assumption seems to me the exact think that is otherwise always recommended: not to make any assumptions; and then, if you make the assumption that one is red and one white, why not also make the assumption that the other is red, and one is white and create the reverse case.

The notion of a Red dice and a White dice is to make the understanding of the problem easier. We are not really making any new assumptions. The two dices are two different entities. If you dont want to assume that they are red and white, then you dont have to.

2- assuming that the dice are rolled in order, which may be true but also not explicitly stated (hence again, assumed) and still, irrelevant for the sum.

You are quire right that the order of rolling the dices are irrelevant to the sum. I dont understand whats your question here.

Anyway, if one dice is red and one white, or one first and one second - the question of why is 2+2 not different than 2+2 is still unanswered.

I don't want to create a controversy here but to clarify this for ever an put it to rest forever. In the grand context of gmat, it's a petty question but I would not like to meet with it on the exam.

Since we all seem to submit to ETS dogma .. did you ever see this question and explanation on an ETS provided material? Thank you!

Lets list all the possibilities of rolling the two dices. The outcomes where the sum is 4 or 6 is bolded.

(1,1), (1,2), (1,3) (1,4), (1,5), (1,6),
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

As you can see (2,2) and (3,3) appear only once.

I hope that answers your question

[tim]

Hi Piccolino,

i hate to have to tell you this, but no one is likely to help you further if you are going to stubbornly reject all the explanations that are given to you. It seems to me that you are having no trouble understanding the explanations, you just don't want to agree with them. Deal with it. If you want to demonstrate something that (you perceive) is WRONG with these explanations, we'll be happy to help. But as things stand, the explanations get you to the correct answer in all cases, so it's up to you now whether you want to get the question right or not..

BTW as for ETS explanations, you're probably better off not even looking for those. We wrote the Official Guide Companion for a reason.. :)

Thanks again for your help, pellucide..

[piccolino]

Oh, I meant to say that if we assume that (1,3) and (3 ,1) are different, then this is equivalent to assume that (2, 2) and (2, 2) are also different (as different colors, or different orders).

But what I am unconvinced is that it matters how the sum is made - for this problem. If it were difference, i could see why it would have mattered.

I appreciate your reply.

[piccolino]

Yes, let's put this to sleep, thank you very much for your help, Tim.

[ChrisB]

Hi,
Thanks,
Chris

[tim]

Piccolino,
Again, i'm happy to help if you are having trouble understanding the explanations, but you're right it's best to lay this one to rest if you understand the given explanations but just don't like the way they feel. Sometimes the techniques just work; it's messy, but they work.. :)

[adm45]

I understand why rolling a 1 and 6 is different from rolling a 6 and 1. Can you explain the underlying concept and how it will be tested on other questions that don't involve a die?

Thanks,

[jlucero]

I understand why rolling a 1 and 6 is different from rolling a 6 and 1. Can you explain the underlying concept and how it will be tested on other questions that don't involve a die?

Thanks,

Anything that has two distinct items or events that could happen in more than a single way. You have better odds flipping one coin two times and getting 1H & 1T than 2T, because there is more than one way to get 1H & 1T. Choosing from a deck of cards one red card and one black card is more likely than choosing two red cards. In both cases, no matter what you get on the first toss/draw, you have the ability to get the other. Getting two red cards requires you to get a red on the first and second draw and two tails requires something happen in flip one and flip two.

[adm45]

can you use a problem to demonstrate how the dice example applies to things you were talking about. thanks! I learn better that way.

[tim]

I am totally confused. You are asking for a problem that demonstrates this principle, but the whole reason we're discussing this principle is because it arose from the original problem in this thread. Am I missing something?

[adm45]

Can you provide example other than dice? You mentioned cards and flipping coins scenario. Can you provide examples them.