The Correct Guess

Similar explanation as @Arindam Kulkarni's, but with more details.

As mentioned by Arindam, the problem is effectively asking how many two relatively prime positive integers or coprime positive integers add up to $504$ . The integers $m$ and $n$ are coprime integers if their greatest common divisor is $1$ or $\gcd(m,n)=1$ . That is $m$ and $n$ don't share any common divisor other than $1$ . To find the number of positive integers that are coprime to $504$ , we use the Euler's totient function $\phi (504)$ which is computed as follows:

$\begin{aligned} \phi (n) & = n \left(1-\frac 1{p_1}\right)\left(1-\frac 1{p_2}\right)\left(1-\frac 1{p_3}\right)\cdots \left(1-\frac 1{p_k}\right) & \small \blue{\text{where }p_1, p_2, p_3, \cdots p_k \text{ are all the prime divisors of }n.} \\ \implies \phi(504) & = 504 \times \frac 12 \times \frac 23 \times \frac 67 = 144 & \small \blue{\text{Since }504 = 2^3 \times 3^3 \times 7} \end{aligned}$

Since Arindam's dog makes $1$ guess out of $144$ possibilities, the probability of guessing the right answer is $\boxed{\frac 1{144}}$ .

The problem is effectively asking how many ways can two relatively prime positive integers add up to 504. If m and n are NOT relatively prime, their GCD can be any factor of 504. This means you want to find how many numbers are relatively prime to 504. This is simply φ(504). This is 504 * $\frac{6}{7}$ * $\frac{1}{2}$ * $\frac{2}{3}$ = 144 .