We throw a six-sided die independently.
Let be $X_1$ a random variable (rv) that counts all throws until we see the the first time two consecutive $6$'s. Let be $X_2$ the rv that ignores the first throw and counts all throws from the second throw until the first time we see two consecutive $6$'s.
Show that the expected values $\mathbb{E}(X_1)$ and $\mathbb{E}(X_2)$ are equal. (Hint: assume that those expected values exist.)
Is it possible to answer this question without using any advanced concepts of measure or probability theory?
Intuitively this statement makes sense. But maybe there is some basic argument I can use to back up my intuition.
My approach:
Let be $\omega$ some element that represents a sequence of dice throws and we assume that $X(\omega)=k$. Though, we don't know how the sample space or the probability measure look like it makes sense that we can identify each sequence of dice throws by $\omega=(\omega_1,\omega_2,\omega_3,\dots)$ and due to the independence of the individual throws the probability measure should satisfy $P((\omega_1,\omega_2,\omega_3,\dots))=p(\omega_1)p(\omega_2)p(\omega_3)\cdots$. Further $\{X=k\}$ represents a set consisting of $\omega$'s where the first two consecutive $6$'s appear at throw $(k-1)$ and $k$.
Now, we simply add another element $\omega_0$ at the beginning of each of those sequences $\omega\in\{X=k\}$, e.g. $(\omega_1,\omega_2,\omega_3,\dots)$ produces six new sequences like $(1,\omega_1,\omega_2,\omega_3,\dots)$, $(2,\omega_1,\omega_2,\omega_3,\dots)$, $(3,\omega_1,\omega_2,\omega_3,\dots)$ etc. As the individual entries $\omega_i$ are independent by assumption this new set, we call it $M$, has the same probability as $\{X=k\}$. By construction all $\omega\in M$ satisfy $X_2(\omega)=k$. Of course we can do it the other way around if we start with the set $\{X_2=k\}$ and cut off the first entry. So we see that $P(\{X_1=k\})=P(\{X_2=k\})$ and as a result both expected values must attain the same value.
Is this a valid reasoning? Maybe someone has another good idea?