Let x denote the number of doublets in three throws of a pair of dice with the following probability distribution.

If value of k is equal to $\frac{m}{n}, gcd(m, n) = 1$, then $m+n$ is equal to

8

The correct answer is Option (1) → 8

Given: Probability distribution of number of doublets in 3 throws of a pair of dice:

Use total probability = 1:

$\frac{25}{72}k + \frac{15}{72}k + \frac{3}{72}k + \frac{1}{360}k = 1$

$k \left( \frac{25 + 15 + 3}{72} + \frac{1}{360} \right) = 1$

$k \left( \frac{43}{72} + \frac{1}{360} \right) = 1$

Take LCM of 72 and 360 = 360

$\frac{43}{72} = \frac{215}{360}$, so:

$k \left( \frac{215 + 1}{360} \right) = 1 \Rightarrow k \cdot \frac{216}{360} = 1$

$k = \frac{360}{216} = \frac{5}{3}$

So $k = \frac{m}{n} = \frac{5}{3} \Rightarrow m + n = {8}$