Coin Probability

Question

Suppose a coin with Pr{H} = 3/4 is flipped 4 times. Answer the following questions: is the probability exactly 2 heads were flipped?

What is the probability that heads then tails then heads then tails was flipped?

Answer 1

A nice tool to use to figure this out is a tree diagram. Unfortunately, I drew one but the file size restrictions of this answer page won't let me paste in in : (.

You would do this, then continue it for the 4 flips:

When you do that, you'll find 6 different combinations that give 2 head and 2 tails. They are HHTT, HTHT, HTTH, THHT, THTH, and TTHH. Each of those has a probability of 9/256 so add those up and you get 54/256. Then the probability of getting HTHT only happens once and the probability is again 9/256