First of all picking two digits is the (10 over 2) as for the first you have 10 options and the second the remaining 9 (And because picking 5,6 or 6,5 is the same).
Now there are two ways to arrange 2 different digits in 4 locations: Either 3,1 or 2,2.
If we're looking at the 3,1 arrangement, pick one of the 2 digits you chose and chose a place out of the 4 to put it.
If we're looking at the 2,2 arrangement, you can pick the smaller of the two digits and pick 2 places out of the 4 to put it in (4 * 3 as you have 4 places to pick for the first copy and another place out of the remaining three). The other digit will fill the remaining two places.
In other words, (10 over 2)*(2*4 + 1*4*3).
The amount of ways to write a 4 digit number out of 10 digits with repetitions is 10^4 as for every place of the 4 you can pick any of the 10 digits.
Resulting in a probability of (10 over 2)*(2*4 + 1*4*3)/10^4.
P.S Both your solution and the other two ignore the fact that 3344 fits the description.
P.P.S In the second case when we said pick the smaller, I didn't bother to explain this in place: If you picked either of the two, you might end up with numbers like 5566 both when picking 5,6 and 6,5 due to symmetry and should therefore divide the result by two. The requirement of picking the smaller one breaks the symmetry so that when 5,6 are present only a single pick is possible, avoiding the double count.