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

JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Probability

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

\frac{p}{q}

, where

p

and

q

are coprime, then the sum of the positive divisors of p is equal to

Answer: (B)

The random variable

X = α - β

takes values from

- 5

5

. By symmetry,

E [X] = 0

. The variance is

Var (X) = E [X^{2}]

. Computing

E [X^{2}]

from the probability distribution (each value

k

occurs with probability

\frac{6 - ∣ k ∣}{36}

for

k \in {- 5, \dots, 5}

E [X^{2}] = 2 k = 1 \sum 5 k^{2} \cdot \frac{6 - k}{36} = \frac{2}{36} (25 \cdot 1 + 16 \cdot 2 + 9 \cdot 3 + 4 \cdot 4 + 1 \cdot 5) = \frac{2}{36} (25 + 32 + 27 + 16 + 5) = \frac{210}{36} = \frac{35}{6} .

p = 35 = 5 \times 7

. The sum of divisors is

(1 + 5) (1 + 7) = 6 \times 8 = 48

JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Single correct option