Find the expected number of times the string "011" will appear in a randomly generated binary string on length 10

Question

What is the number of times the string "011" will appear as a substring in a randomly generated binary string of length 10. All the bits of the random binary string have equal probability of being 0 or 1. My solution uses indicator random variables. Let us have an indicator random variable $X_i$, such that $X_i = 1$ if $011$ occurs at $ith$ position in the random binary string and $X_i = 0$ otherwise. Then $E[Xi] = P(Xi) = 1/8$ (There are 8 binary strings of length 3. Favorable case is string $011$) Now we want to calculate expected value of number of appearances of $011$, call this $E[Z]$. $E[Z] = E[X1] + E[X2] + E[X3] + ... + E[X8]$ (011 can not start from index 9 or 10 in a string of length 10). So $E[Z] = 8 * 1/8 = 1$ Is this solution correct?

Answer 1

Yes, that looks correct. Additivity of expectations helps you avoid what could otherwise have been a rather complex calculation.