Question

Two dice A and B are rolled, Let the numbers obtained on A and B be α and β respectively. If the variance of α - β is \(\frac{p}{q}\), where p and q are coprime, then the sum of the positive divisors of p is equal to

• A. 31
• B. 36
• C. 48
• D. 72

Answer 1

The Correct Option is C

To determine the variance of \( \alpha - \beta \), where \(\alpha\) and \(\beta\) are the numbers obtained when two dice A and B are rolled, we start by calculating the variance in a step-by-step manner.

1. The expected value of \(\alpha\) (expected value of a single die roll): E(\alpha) = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5
2. Similarly, for \(\beta\): E(\beta) = 3.5
3. Variance of \(\alpha\) (or a single die) is: \mathrm{Var}(\alpha) = E(\alpha^2) - (E(\alpha))^2
4. Calculate \(E(\alpha^2)\): E(\alpha^2) = \frac{1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2}{6} = \frac{91}{6}\
5. Thus, variance of \(\alpha\): \mathrm{Var}(\alpha) = \frac{91}{6} - 3.5^2 = \frac{35}{12}
6. The variance of \(\beta\) is the same as \(\alpha\), i.e., \( \frac{35}{12} \).
7. Variance of \(\alpha - \beta\) is given by: \mathrm{Var}(\alpha - \beta) = \mathrm{Var}(\alpha) + \mathrm{Var}(\beta) - 2\mathrm{Cov}(\alpha, \beta)\
8. Since \(\alpha\) and \(\beta\) are independent, \(\mathrm{Cov}(\alpha, \beta) = 0\). Thus: \mathrm{Var}(\alpha - \beta) = \mathrm{Var}(\alpha) + \mathrm{Var}(\beta) = \frac{35}{12} + \frac{35}{12} = \frac{70}{12} = \frac{35}{6}\
9. The fraction \(\frac{35}{6}\) is in its simplest form with coprime numbers p and q where p = 35 and q = 6.

The sum of the positive divisors of \( p = 35 \) is calculated as follows:

• The divisors of 35 are 1, 5, 7, and 35.
• Therefore, the sum of these divisors is \(1 + 5 + 7 + 35 = 48\).

Thus, the sum of the positive divisors of \( p \) is 48, which matches the provided correct answer.