I built the probability tree using PowerPoint. It's not freeware, but it's easy to learn and use. If you don't have PowerPoint you can try Google Docs, but I don't know if they have the same widgets that PowerPoint has or not.
I've been following this thread without comment as there wasn't anything I could contribute (didn't have the aha! moment like harey did). I'm convinced that harey's approach is correct. Probably the easiest way to see it is to draw the probability tree of complementary events (see below). The green track leads to the complete annihilation with total probability established earlier, the red track is the "no collision" track with total probability of 1/n! and the yellow tracks are all the partial annihilation tracks. This tree also explains jasen's finding that the most likely outcome is that 2 bullets remain. Half the leaf nodes of this tree will have that outcome and the total probability will be the sum of those leaf node's probabilities.
I would agree with jasen's answer. I think you're reading too much into details that are merely artifacts of the quality of the test. In this second puzzle there are 9 ways to combine 2 sticks of 3 types. 8 of them are given and the 9th is missing.
Well, I don't have the context that you have as I don't see the rest of the questions, but looking at this question alone I wouldn't call it particularly complex and my solution/answer seems reasonable to me. BTW, I've never seen a test where two of the possible answers were exactly the same as in this test.
Given that John and Julie play on the same level and have 50/50 chance of winning any particular game, I agree with Bonanova that Julie has 50% chance of winning whatever series they choose to play. However, I think the OP is asking about the probability of Julie winning with the score 4:3 in the 7 game series vs. the probability of winning 5:4 in the 9 game series. Those probabilities are different...