Coin Toss, Counts Expectation of Three Head In A Row

This a phone interview question from Morgan Stanley in 2010.

We continue toss one coin until it appears three ( $x$ ) heads in a row, then stops. Let the number of toss be $Y$. What is the expectation of  $Y$? 

Let's start with the simple case, $ x = 1 $. We have half chance to get head (H). If we get tail (T), we back to the beginning and start over again.

$$E(Y) = 1 \times 0.5 + (E(Y) + 1) \times 0.5$$
Solution: $E(Y) = 2$.

Two key points:
  • $E(Y)$ need involved when toss game start over.
  • Don't forget 1 in $(E(Y) + 1)$, since there is one toss taken to get here.
For $ x = 2 $,  the whole probability space are T, HT, HH. $E(Y) = 6$.
For $ x = 3 $,  the whole probability space are T, HT, HHT, HHH. $E(Y) = 14$.

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