When rolling a certain unfair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6, the probability of obtaining face $F$ is greater than $1/6$, the probability of obtaining the face opposite face $F$ is less than $1/6$, the probability of obtaining each of the other faces is $1/6$, and the sum of the numbers on each pair of opposite faces is 7. When two such dice are rolled, the probability of obtaining a sum of 7 is 47/288. Given that the probability of obtaining face $F$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Guest Apr 8, 2015

#2**+5 **

Hi Mellie,

It is very late and I am not going to get a final answer tonight but I will show you where my thoughts are taking me.

I'm going to let 1 and 6 be the biased sides so 2,3,4 and 5 all have a chance of 1/6 of being rolled.

so the chance of throwing a 1,2,3 or 4 is $${\frac{{\mathtt{4}}}{{\mathtt{6}}}}$$ = $${\frac{{\mathtt{2}}}{{\mathtt{3}}}}$$

the chance of throwing a 1 is $${\frac{{\mathtt{m}}}{{\mathtt{n}}}}$$ and the chance of throwing a 6 is $${\frac{{\mathtt{1}}}{{\mathtt{3}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{m}}}{{\mathtt{n}}}}$$

now to roll a 7 these are the possibilities

$$\\P(1,6) =\frac{m}{n}\times\frac{n-3m}{3n}=\frac{m(n-3m)}{3n^2}\\\\

P(2,5) = \frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\\\\

P(3,4) = \frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\\\\

P(4,3) = \frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\\\\

P(5,2) = \frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\\\\

P(6,1) = \frac{n-3m}{3n\times}\frac{m}{n}=\frac{m(n-3m)}{3n^2}\\\\

NOW\\\\$$

$$\\\frac{4}{36}+\frac{2m(n-3m)}{3n^2}=\frac{47}{288}\\\\

\frac{32}{288}+\frac{2m(n-3m)}{3n^2}=\frac{47}{288}\\\\

\frac{2m(n-3m)}{3n^2}=\frac{15}{288}\\\\

\frac{m(n-3m)}{n^2}=\frac{15*3}{288*2}\\\\

\frac{m(n-3m)}{n^2}=\frac{5}{64}\\\\

$Now if we are lucky n will equal 8$\\\\

m(8-3m)=5\\\\

8m-3m^2=5\\\\

$Just by sight I can see that m=1 will work.$\\\\

$So one possibility, maybe the only possibility is $ n=8 \;and\; m=1\\\\

m+n=9$$

There is probably a much quicker way of doing this. Plus I have not shown that this is the ONLY solutions.

Melody Apr 9, 2015

#2**+5 **

Best Answer

Hi Mellie,

It is very late and I am not going to get a final answer tonight but I will show you where my thoughts are taking me.

I'm going to let 1 and 6 be the biased sides so 2,3,4 and 5 all have a chance of 1/6 of being rolled.

so the chance of throwing a 1,2,3 or 4 is $${\frac{{\mathtt{4}}}{{\mathtt{6}}}}$$ = $${\frac{{\mathtt{2}}}{{\mathtt{3}}}}$$

the chance of throwing a 1 is $${\frac{{\mathtt{m}}}{{\mathtt{n}}}}$$ and the chance of throwing a 6 is $${\frac{{\mathtt{1}}}{{\mathtt{3}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{m}}}{{\mathtt{n}}}}$$

now to roll a 7 these are the possibilities

$$\\P(1,6) =\frac{m}{n}\times\frac{n-3m}{3n}=\frac{m(n-3m)}{3n^2}\\\\

P(2,5) = \frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\\\\

P(3,4) = \frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\\\\

P(4,3) = \frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\\\\

P(5,2) = \frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\\\\

P(6,1) = \frac{n-3m}{3n\times}\frac{m}{n}=\frac{m(n-3m)}{3n^2}\\\\

NOW\\\\$$

$$\\\frac{4}{36}+\frac{2m(n-3m)}{3n^2}=\frac{47}{288}\\\\

\frac{32}{288}+\frac{2m(n-3m)}{3n^2}=\frac{47}{288}\\\\

\frac{2m(n-3m)}{3n^2}=\frac{15}{288}\\\\

\frac{m(n-3m)}{n^2}=\frac{15*3}{288*2}\\\\

\frac{m(n-3m)}{n^2}=\frac{5}{64}\\\\

$Now if we are lucky n will equal 8$\\\\

m(8-3m)=5\\\\

8m-3m^2=5\\\\

$Just by sight I can see that m=1 will work.$\\\\

$So one possibility, maybe the only possibility is $ n=8 \;and\; m=1\\\\

m+n=9$$

There is probably a much quicker way of doing this. Plus I have not shown that this is the ONLY solutions.

Melody Apr 9, 2015