Two numbers are chosen independently and uniformly at random from the set $\{1, 2, \dots, 13\}$. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary representations have the same most significant bit is _______.

The set of numbers is $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}$, so $|S| = 13$.

In 4-bit unsigned binary representation:

• Numbers with most significant bit (MSB) equal to $0$ are $1$ to $7$ (since $1_{10} = 0001_2$ to $7_{10} = 0111_2$). There are $7$ such numbers.

• Numbers with MSB equal to $1$ are $8$ to $13$ (since $8_{10} = 1000_2$ to $13_{10} = 1101_2$). There are $6$ such numbers.

Two numbers are chosen independently and uniformly with replacement from the set.

• Probability that both numbers have MSB = $0$:

• Probability that both numbers have MSB = $1$:

• Total probability of having the same MSB:

Rounding off to 3 decimal places gives 0.503 (acceptable range: 0.502 to 0.504).