The second number in the given number-pairs is obtained by performing certain mathematical operation(s) on the first number. The same operation(s) is/are followed in all the number-pairs except one. Find that odd number-pair.

To find the odd number-pair, let us analyze the mathematical relationship between the first number ($x$) and the second number ($y$) in each pair.

By observing the numbers, we can see a common pattern where the second number is obtained by dividing the first number by 2 and then adding 1:

$y=\frac{x}{2}+1$

Let us check this relation for each of the given pairs:

1. For the pair 20 : 11:

$\frac{20}{2}+1=10+1=11$
(This pair follows the pattern.)

2. For the pair 6 : 4:

$\frac{6}{2}+1=3+1=4$
(This pair follows the pattern.)

3. For the pair 12 : 7:

$\frac{12}{2}+1=6+1=7$
(This pair follows the pattern.)

4. For the pair 15 : 9:

$\frac{15}{2}+1=7.5+1=8.5\ne 9$
(This pair does not follow the pattern.)

Since the pair 15 : 9 does not satisfy the mathematical logic shared by the other three options, it is the odd number-pair.