I am currently studying probability, and I've gotten pretty far (expectation conditioning, moment generating functions, etc.) and I can understand those concepts pretty well, but I am struggling with creating random variables. For example, I see a basic question like
Consider n independent flips of a coin having probability p of landing heads. Say a changeover occurs whenever an outcome differs from the one preceding it. For instance, if the results of the flips are HHTHTHHT , then there are a total of five changeovers. If p = 1/2, what is the probability there are k changeovers?
or
Suppose that two teams are playing a series of games, each of which is independently won by team A with probability p and by team B with probability 1 − p. The winner of the series is the first team to win four games. Find the expected number of games that are played, and evaluate this quantity when p = 1/2
and I have absolutely no idea how to formulate the random variables in order to begin solving the question.
I was working on exercises related to the higher level concepts, and I found that if the variables have already been formulated for me, or if it is was a question which does not require formulating of random variables, I have a much easier time solving them. I have tried to practice a bunch of questions, and currently I've reverted to much simpler questions (like the ones above) but it still does not make any sense. It always seems like there is a new trick to making the variables. Can anyone help me develop a better methodology for formulating random variables from questions?
To be clear, I am not looking for solutions to the specific questions above, but a better methodology to solve them.
Apologies if this question does not apply to this forum, but any help would be appreciated!