1. Define the random variables

Let $\alpha$ and $\beta$ be the outcomes on two fair dice $A$ and $B$.

Each die is independent and uniformly distributed over $\{1,2,3,4,5,6\}$.

We need: $\operatorname{Var}(\alpha-\beta)$

2. Use variance properties

Since $\alpha$ and $\beta$ are independent, $\operatorname{Var}(\alpha-\beta)=\operatorname{Var}(\alpha)+\operatorname{Var}(\beta)-2\operatorname{Cov}(\alpha,\beta)$ But independence gives $\operatorname{Cov}(\alpha,\beta)=0$ So, $\operatorname{Var}(\alpha-\beta)=\operatorname{Var}(\alpha)+\operatorname{Var}(\beta)=2\operatorname{Var}(\alpha)$

3. Find variance of one fair die

For a fair die $X$ taking values $1,2,3,4,5,6$:

$E[X]=\frac{1+2+3+4+5+6}{6}=\frac{21}{6}=\frac{7}{2}$

Now, $E[X^2]=\frac{1^2+2^2+3^2+4^2+5^2+6^2}{6}$ $=\frac{1+4+9+16+25+36}{6}=\frac{91}{6}$

Hence, $\operatorname{Var}(X)=E[X^2]-(E[X])^2$ $=\frac{91}{6}-\left(\frac{7}{2}\right)^2$ $=\frac{91}{6}-\frac{49}{4}$ Taking LCM $12$, $\operatorname{Var}(X)=\frac{182-147}{12}=\frac{35}{12}$

Therefore, $\operatorname{Var}(\alpha-\beta)=2\cdot \frac{35}{12}=\frac{35}{6}$

So, $\frac{p}{q}=\frac{35}{6}$ with $p=35$, $q=6$.

4. Find sum of positive divisors of $p$

The positive divisors of $35$ are: $1,5,7,35$ Their sum is $1+5+7+35=48$

5. Evaluate options

• A: $48$ ✅
• B: $31$ ❌
• C: $72$ ❌
• D: $36$ ❌

Therefore, the correct option is A.