# independent flips of a coin

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?

 ebaines Posts: 10,576, Reputation: 5794 Expert #2 Sep 21, 2007, 08:31 AM
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?

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