Independent flips of a coin that lands on heads with probability p are made. What is the probability that the first four outcomes are
A: H,H,H,H?
B: T,H,H,H?
C: what is the probability that the pattern T,H,H,H occurs before the pattern H,H,H,H?

I'll get you started with the first problem, then let us know whether this helps you get through the others.

With each flip of a fair coin, the probability of getting a heads is 1/2. Now think about the probability of getting two heads in a row - half the time you'll get that first heads, and then if that is successful half the time you'll get a second head. So the probability of getting two heads in a row is 1/2 * 1/2, or 1/4. Another way to think of this is that for two flips of a coin there are 4 possible outcomes:
HH
HT
TH
TT
All of these outcomes are equally likely, and only one satisfies the condition of 2 heads in a row; consequently the probability of getting two heads in a row is 1 out of 4 or 1/4.

Now, can you extend this line of reasoning to determine the probability of getting 4 heads in a row? How about the probability of 1 tail followed by 3 heads?