# Math Help - Dice Probability in an Excel Spreadsheet

1. ## Dice Probability in an Excel Spreadsheet

A person throws a normal 6-sided dice a certian number of times, and their score is the highest value they throw. What is the probability of them getting a certain score?

I need to calculate the probability of getting all the scores 1 to 6, for up to about 5 throws. I can of course do this manually in a spreadsheet by entering an individual formula into every cell, but is there a more general formula I could use? I've tried looking at my notes but although I generally understand probabilities and permutations/combinations, I'm pretty stuck...

Any help would be really appreciated!

2. ## Re: Dice Probability in an Excel Spreadsheet

- calculate the probability to have no 6.
-- this gives you the probability to have at least one 6.
- calculate the probability to have no 6 and no 5.
-- the difference between both gives you the probability to have at least one 5, but no 6.
continue the pattern for all other values.

3. ## Re: Dice Probability in an Excel Spreadsheet

Originally Posted by BeccyM
A person throws a normal 6-sided dice a certian number of times, and their score is the highest value they throw. What is the probability of them getting a certain score? I need to calculate the probability of getting all the scores 1 to 6, for up to about 5 throws.
I can of course do this manually in a spreadsheet by entering an individual formula into every cell, but is there a more general formula I could use?
Frankly I puzzled by the statement in blue. Because of the particular rules of this game, I doubt there is a neatly compact solution. This is a counting problem. As such this solution may not be any different from what you have in mind with a spreadsheet.

Say we roll the die $N$ times. Let $k=1,~2,\cdots,~6$.
Define a finite sequence $d_1=1$ and for $k\ge 2,~d_k={k^N} - \sum\limits_{j = 1}^{k - 1} {{d_j}}$.
That is a sequence of the number of ways of winning for each $k$.

So for example, the probability of four winning is $\frac{d_4}{6^N}$.

4. ## Re: Dice Probability in an Excel Spreadsheet

Here is another approach.

Let's say the die is rolled n times and the maximum number is X. Consider a single roll; the probability that the score is less than or equal to x is $x / 6$. So the probability that all n rolls are less than or equal to x is $(x / 6)^n$, i.e.
$\Pr(X \leq x) = (x / 6)^n$.

Now you can use $\Pr(X = x) = \Pr(X \le x) - \Pr(X \le x-1)$.

5. ## Re: Dice Probability in an Excel Spreadsheet

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6. ## Re: Dice Probability in an Excel Spreadsheet

Think of it this way: If X is the score you obtain, then

$P(X=1) = (\frac{1}{6})^5$, i.e. you need to roll a 6 five times in a row to score 1.

To score 2, you need to roll a 1 or 2 on each die, but at least one die must be a 2. It follows that

$P(X=2) = (\frac{2}{6})^5 - (\frac{1}{6})^5$

This generalizes to

$P(X=k) = (\frac{k}{6})^5 - (\frac{k-1}{6})^5 - \cdots - (\frac{1}{6})^5$

And obviously the 5 can be replaced with any other positive integer, in order to generalize even more.

7. ## Re: Dice Probability in an Excel Spreadsheet

Thank you everyone for your help!

To: mfb,

Yes I think it is something like that, but I can't think of a formula that I could copy and paste into every cell lol!

To: Plato,

So for a score of 4 in 1 throw, the formula would be

(4^1 - (3^1 + 2^1 + 1^1) / (6^1) ???

And of course you are right about the fact that neat compact solution is what I'm looking for anyway :/

To: awkward, richard1234

You seem to be saying that if "n" is the no. of throws, the probability of getting a score of 1 is (1/6)^n, and then the probability of getting a 2 is (2/6)^n - (1/6)^n and so on (if I understand correctly)

However things seem to go horribly wrong when I enter this formula into a spreadsheet :/

For example, the probability of getting a score of 3 with 1 throw is, according to this formula, (3/6)^1 - ((2/6)^1 + (1/6)^1)) = 0 (!)

Also, when I repeat the calculation with other values the probabilities don't add up to 1... I think it might be to do with the fact that there are lots of different permutations of the different scores?

*Edit* I think I might've worked it out... Very complicated it took me an hour... If anyone could check and confirm if you think I've got it right that would be much appreciated!

Where n = no. of throws, s = score

P(S) = Q - (the sum of all the probabilities of the scores greater than that score)

where Q is a cumulative distribution function with the following parameters:

Number of successes in trials = (n-1)
No. of trials = n
Probability of success = s/6

My apologies if this isn't explained very clearly. Here's the spreadsheet:

(I think you have to go to file>download to see the formulae)

8. ## Re: Dice Probability in an Excel Spreadsheet

Originally Posted by BeccyM
To: Plato,
So for a score of 4 in 1 throw, the formula would be
(4^1 - (3^1 + 2^1 + 1^1) / (6^1) ???
I have attached a image of the probabilities for 1 to 6 in five throws.

BTW. Reply #4 is exactly the same as mine except I actually computed the probabilities.