Stupid question in probability that amazed me

Dear all,

    Here is a simple question of probability that I tried hard for a half
day with no result.

Given a game of tossing coin, not until the recent three tosses have exactly
two heads up, I will keep tossing.
What is the expectation of the number of toss I have the made?

    The problem looks simple, but it turns out quite hard to solve, at
least, for me. Please spend a little time to try it out ^^

Thanks for reading my message!

Andrew Au

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