Hi. Is there a way to solve the following without actually listing out all the possibilities?

What is the probability of rolling a total sum of 8 using three dice?

Thank you! :)

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- November 27th 2006, 05:04 PMfeiyingxDice Probability
Hi. Is there a way to solve the following without actually listing out all the possibilities?

What is the probability of rolling a total sum of 8 using three dice?

Thank you! :) - November 27th 2006, 06:44 PMgalactus
Yes, there is a way.

There are 3 dice and you want to find all the sums of 8.

Expand this out and check the coefficient of

That is the number of ways to sum to 8.

The coefficent is

Therefore, there are 21 ways.

There are possible rolls with 3 dice.

Probability is - November 27th 2006, 09:00 PMfeiyingx
Wow! Ingenious xD.

Now can u explain why that is? I don't see it.

Thanks again! - November 28th 2006, 03:52 AMgalactus
A text on advanced counting methods and/or generating functions may prove useful if you are curious of how it's derived.

- November 28th 2006, 08:51 AMfeiyingx
Thank you. =]

- December 18th 2006, 06:42 PMF.A.P
This is a more direct approach for those not familiar with generating functions.

Let Z be the sum of three dice. Let Y be the sum of the first two dice.

Now

, for k = 2, 3,...,7, and 0 elsewhere. Also it's easy to show that

, for k = 2, 3,...,7 (we won't consider higher k's)

Hence we end up with