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-   -   3 dice probabilities (http://mymathforum.com/probability-statistics/339330-3-dice-probabilities.html)

newbieDes March 4th, 2017 08:59 AM

3 dice probabilities
 
Here are the well known probability results for rolling two dice.

2 -2.8%
3- 5.6
4 - 8.3
5- 11.1
6- 13.9
7 - 16.7
8 - 13.9
9 - 11.1
10 - 8.3
11 - 5.6
12 - 2.8

Can someone provide similar results for rolling 3 dice that includes summing 2 of the dice together? Example: rolling a 3, 4 and 5 could be 7 and 5, 9 and 3, or 8 and 4.

Thanks in advance.

MarkFL March 4th, 2017 11:16 AM

What I would do is first observe that the sums $S$ will range from 3 to 18, and the total number of outcomes is $N=6^3=216$. We should also expect the probabilities to be symmetric about $S=10.5$.

And so we begin with $S=3$. There's only 1 way (the permutations of (1,1,1)) to get 3, so the probability of rolling a sum of 3 is:

$\displaystyle P(3)=\frac{1}{216}$

For $S=4$, we find there are 3 permutations of (1,1,2), hence:

$\displaystyle P(4)=\frac{3}{216}=\frac{1}{36}$

For $S=5$, we find there are 3 permutations of (1,2,2) and 3 permutations of (1,1,3), hence:

$\displaystyle P(5)=\frac{3+3}{216}=\frac{1}{18}$

For $S=6$, we find 3 permutations of (1,1,4), 6 permutations of (1,2,3), and 1 permutation of (2,2,2), hence:

$\displaystyle P(6)=\frac{3+6+1}{216}=\frac{5}{108}$

For $S=7$, we find 3 permutations of (1,1,5), 6 permutations of (1,2,4), 3 permutations of (1,3,3), and 3 permutations of (2,2,3), hence:

$\displaystyle P(7)=\frac{3+6+3+3}{216}=\frac{5}{72}$

For $S=8$, we find 3 permutations of (1,1,6), 6 permutations of (1,2,5), 6 permutations of (1,3,4), 3 permutations of (2,2,4), 3 permutations of (2,3,3), hence:

$\displaystyle P(8 )=\frac{3+6+6+3+3}{216}=\frac{7}{72}$

For $S=9$, we find 6 permutations of (1,2,6), 6 permutations of (1,3,5), 3 permutations (1,4,4), 3 permutations of (2,2,5), 6 permutations of (2,3,4) and 1 permutation of (3,3,3), hence:

$\displaystyle P(9)=\frac{6+6+3+3+6+1}{216}=\frac{25}{216}$

For $S=10$, we find 6 permutations of (1,3,6), 6 permutations of (1,4,5), 3 permutations of (2,2,6), 6 permutations of (2,3,5), 3 permutations of (2,4,4), and 3 permutations of (3,3,4), hence:

$\displaystyle P(10)=\frac{6+6+3+6+3+3}{216}=\frac{1}{8}$

Now we can use symmetry to fill in the table:

$\displaystyle \newcommand\T{\Rule{0pt}{1em}{1em}}
\begin{array}{|c|c|}
\hline S & P(S) \T \\\hline
3 \T & \dfrac{1}{216} \\\hline
4 \T & \dfrac{1}{36} \\\hline
5 \T & \dfrac{1}{18} \\\hline
6 \T & \dfrac{5}{108} \\\hline
7 \T & \dfrac{5}{72} \\\hline
8 \T & \dfrac{7}{72} \\\hline
9 \T & \dfrac{25}{216} \\\hline
10 \T & \dfrac{1}{8} \\\hline
11 \T & \dfrac{1}{8} \\\hline
12 \T & \dfrac{25}{216} \\\hline
13 \T & \dfrac{7}{72} \\\hline
14 \T & \dfrac{5}{72} \\\hline
15 \T & \dfrac{5}{108} \\\hline
16 \T & \dfrac{1}{18} \\\hline
17 \T & \dfrac{1}{36} \\\hline
18 \T & \dfrac{1}{216} \\\hline
\end{array}$

newbieDes March 4th, 2017 03:52 PM

Mark, thanks for the reply.

What I wanted to calculate is the odds of rolling 1-12 using three dice but locking two of them (2-12) and using the third one by itself (1-6).

Since I mentioned every roll would include locking two dice, the value of the locked pair would never exceed 12 and the value of the single dice would never exceed 6. I don't want to calculate the sum of these two values but want to know the odds of how many times 1-12 will be rolled when each roll involves locking two dice and using one by itself. Hope that is more clear than my original post.

JeffM1 March 4th, 2017 04:27 PM

I am confident that there is a way to calculate whatever probabilities you want provided that we understand what you want. I must admit that I do not want understand.

MarkFL March 4th, 2017 05:02 PM

Quote:

Originally Posted by JeffM1 (Post 563543)
I am confident that there is a way to calculate whatever probabilities you want provided that we understand what you want. I must admit that I do not want understand.

Yes, my initial post we made based on my best guess as to what was actually being asked...and I have to admit I still don't understand either. :confused:

fungarwai March 5th, 2017 02:43 AM

We have a generating function for
rolling n dice

$\displaystyle (x^1+x^2+x^3+x^4+x^5+x^6)^n$

$\displaystyle (x^1+x^2+x^3+x^4+x^5+x^6)^2=x^2+2x^3+3x^4+4x^5+5x^ 6+6x^7+5x^8+4x^9+3x^{10}+2x^{11}+x^{12}$

observe the coefficient of each $\displaystyle x^m$,you will get the number of ways to get the sum of m for
rolling n dice

$\displaystyle (x^1+x^2+x^3+x^4+x^5+x^6)^n=\frac{x^n(1-x^6)^n}{(1-x)^n}=x^n(\sum_{r=0}^n C_n^r (-1)^r x^{6r})(\sum_{k=0}^\infty C_{n-1+k}^k x^k)$

when n=3,m=4

$\displaystyle x^3(\sum_{r=0}^3 C_3^r (-1)^r x^{6r})(\sum_{k=0}^\infty C_{2+k}^k x^k)$

look for r=0,k=1

$\displaystyle C_{2+1}^1=3$

there are 3 ways to get 4,which are "1+1+2","1+2+1","2+1+1"

when n=3,m=18

$\displaystyle x^3(\sum_{r=0}^3 C_3^r (-1)^r x^{6r})(\sum_{k=0}^\infty C_{2+k}^k x^k)$

look for r=0,1,2 and k=15,9,3

$\displaystyle C_{2+15}^{15}-3C_{2+9}^9+3C_{2+3}^3=1$

newbieDes March 5th, 2017 06:20 AM

Let’s try this.

Bob goes to the local 12 theatre Cineplex every day and chooses two movies to see based on the results of rolling three dice. One dice is chosen to determine his first theatre choice, and the other two dice are added together to determine his 2nd theatre choice.

Example: Bob rolls three dice and can choose to see two movies in these theatres:

6,6,6= Theatre 12 & 6
1,1,1= Theatre 1 and 2
2,4,5= Theatre 2,4,5,6,7,9
3,4,5= Theatre 3,4,5,7,8,9
4,4,6= Theatre 4,6,8,10

Question- How often will Bob be able to see a movie in theatre 1? Theatre2? Etc.

EvanJ March 7th, 2017 04:07 PM

Here are the probabilities based on a spreadsheet I made (hopefully without any mistakes):

1: 91/216
2: 104/216
3: 115/216
4: 128:216
5: 139/216
6: 152/216
7: 92/216
8: 75/216
9: 61/216
10: 46/216
11: 30/216
12: 16/216
Sum: 1,049/216

newbieDes March 7th, 2017 05:23 PM

Quote:

Originally Posted by newbieDes (Post 563584)
Let’s try this.

Bob goes to the local 12 theatre Cineplex every day and chooses two movies to see based on the results of rolling three dice. One dice is chosen to determine his first theatre choice, and the other two dice are added together to determine his 2nd theatre choice.

Example: Bob rolls three dice and can choose to see two movies in these theatres:

6,6,6= Theatre 12 & 6
1,1,1= Theatre 1 and 2
2,4,5= Theatre 2,4,5,6,7,9
3,4,5= Theatre 3,4,5,7,8,9
4,4,6= Theatre 4,6,8,10

Question- How often will Bob be able to see a movie in theatre 1? Theatre2? Etc.

Someone posted this on another forum:

"There are four variables here: A, B and C are dice (where Nature chooses a value of 1 to 6 for each) and D (the player's decision of which die to stand alone). This gives you 6x6x6x3 = 648 total states."


1 36 00 36 / 216 = 16.67%
2 36 06 42 / 216 = 19.44%
3 36 12 48 / 216 = 22.22%
4 36 18 54 / 216 = 25.00%
5 36 24 60 / 216 = 27.78%
6 36 30 66 / 216 = 30.56%
7 00 36 36 / 216 = 16.67%
8 00 30 30 / 216 = 13.89%
9 00 24 24 / 216 = 11.11%
10 00 18 18 / 216 = 08.33%
11 00 12 12 / 216 = 05.56%
12 00 06 06 / 216 = 02.78%


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