The n stands for the number of turns, ie the nth turn.
In order for judy to win the score must be 4 or more. For your table in part ii, you should've got the probability of the score being 4 or more being equal to 0.4 Now, before tackling this lets look at an easier example, calculating the probability that she wins within two turns.
For this to occur the first round had to have been a draw and the score of the second round has to be 4 or more. Calculating the probability of this:
p=0.2(probability that score is 0, ie its a draw) multiplied by 0.4 (probability that score is 4 or more)
Lets do this again to calculate the probability that she wins within three turns this time, we'll be using the same tactic.
For this to happen, the first two rounds had to have been a draw, and, using our formula above we get:
Probability that she wins within three turns= 0.2x0.2x0.4, or (0.2)squared times 0.4. Notice that the power of 0.2 is always one less than the turn number, hence the general formual for Judy winning in the nth turn = (0.2) to the power of n-1 times 0.4.
We have the answer...