In regards to:
https://math.stackexchange.com/a/1584597/323580
Question:
I flip a fair coin independently 6 times. This gives me a sequence of heads and tails. For each consecutive 'HTH' in the sequence you win $5.
So I define a random variable X to be the amount of dollars that I win. For example here is what $X=5$ is: $THTHTT$, and $X=10$ is: $THTHTH$. What is the best way to calculate the expected value of X?
Accepted Answer:
We use the method of indicator random variables, and solve the problem for $n$ tosses. For $k=3$ to $n$, let random variable $X_k$ be $1$ if there is an HTH sequence ending at the $k$-th toss, and let $X_k=0$ otherwise. Then the amount $W$ that we win is given by
$$W=5(X_3+X_4+\cdots +X_n).$$
By the linearity of expectation we have
$$E(W)=5E(X_3)+5E(X_4)+\cdots+5E(X_n).$$
We have $E(X_i)=\Pr(X_i=1)=\frac{1}{8}$, so $E(W)=\dfrac{5(n-2)}{8}$.Only minor modification is needed for a biased coin that has probability $p$ of landing heads.
André Nicolas (https://math.stackexchange.com/users/6312/andr%c3%a9-nicolas), Expected value of coin flip sequence, URL (version: 2015-12-21): https://math.stackexchange.com/q/1584597
This user has stated on his profile that he cannot answer questions about past posts:
I am wondering why $Pr(X_i=1) = 1/8$ ? Since there are $6$ coin flips, there are $2^6$ possible outcomes. Did he calculate every single way there can be a sequence of $HTH$ and divided by $2^6$ to get $1/8$ or is there an easier way? I believe there is an easier, more intuitive way, but I don't know how.
What if I decided I wanted my indicator random variable to be $1$ if $X_i, X_{i+1}, X_{i+2}$ creates the sequence $HTH$ respectively ($k = 1$ to $4$ inclusive)?
Best Answer
Basically, it is simply
$\frac18$ of the sequences will be $HTH?????$ so $P(X_3=1)=\frac18$
$\frac18$ of the sequences will be $?HTH????$ so $P(X_4=1)=\frac18$
$\frac18$ of the sequences will be $??HTH???$ so $P(X_5=1)=\frac18$
$\frac18$ of the sequences will be $???HTH??$ so $P(X_6=1)=\frac18$
$\frac18$ of the sequences will be $????HTH?$ so $P(X_7=1)=\frac18$
$\frac18$ of the sequences will be $?????HTH$ so $P(X_8=1)=\frac18$
Your suggested indexing simply lowers the indices by $2$.