Your assumptions about the game rules are correct. I don't understand a couple of things (sorry really bad at probability).

1. The answers are

and

for (i) and (ii) respectively.

2. When you get 0.32 for the probability of (ii) why do you multiply by 2? I think it should be 0.16. This affects the ratio

. Then the probability for (ii) is

3. How do you get 1/9 for (i)

Here is how I've solved it.

see my first post.

Analogically

My idea was to I calculate the probability they achieve (i) or (ii) after x misses. Then I had to sum the probabilities for x=0,1,2,...