TITLE: Looking for intuition behind coin-flipping pattern expectation
QUESTION [8 upvotes]: I was discussing the following problem with my son:
Suppose we start flipping a (fair) coin, and write down the sequence; for example it might come out HTTHTHHTTTTH....  I am interested in the expected number of flips to obtain a given pattern.  For example, it takes an expected 30 flips to get HHHH.  But here's the (somewhat surprising) thing: it takes only 20 expected flips to get HTHT.  
The tempting intuition is to think that any pattern XXXX is equiprobable since, in batches of 4 isolated flips, this is true.  But when we are looking for embedded patterns like this, things change.  My son wanted to know why HTHT was so much more likely to occur before HHHH but I could not articulate any kind of satisfying explanation.  Can you?

REPLY [6 votes]: The other answers are perfectly good, but I feel a picture is worth a thousand words in this case.

When waiting for HHHH, you go directly to jail (do not pass go, do not collect $200) a lot more often than when you are waiting for HTHT.