**Quote:**

then the probability that someone fills position 1 from Party 1 is 1/2 (because it doesn't matter who fills the position as long as it is someone from Party 1)

Here is the moment in your reasoning that led to the wrong answer. You have calculated the probability for the specific party (Party 1). While the desired event is filling two seats with representatives from the same party. They both can be either from Party 1 or from Party 2.

Thus when you add the probability that the both seats go to Party 2 to your answer, you'll get the correct result:

12/49 + 12/49 = 24/49

**Quote:**

If you relate this to the probability of flipping heads twice, it is 1/2 x 1/2 = 1/4 and 12/49 ~ 1/4

Similarly, you should compare not to "flipping heads twice", but to "flipping one side twice". This side can be either heads or tails.