How many 1's are we expected to get in dice generated sequence?

Question

We are throwing a regular dice until we get a sequence of 1,2,3. What is expected amount of 1's in the result of our dice rolling experiment ?

Answer 1

Let $X_0$ be the total number of 1s. Let $X_{1}$ be the total number of $1s$ after a $1.$ Let $X_{12}$ be tot total number of 1s after a $12.$ Then:

$$\begin{align} E(X_0)&=\frac16(E(X_1)+1)+\frac56 E(X_0)\\ E(X_1)&=\frac16(E(X_1)+1)+\frac16E(X_{12})+\frac46E(X_0)\\ E(X_{12})&=\frac16(E(X_1)+1)+\frac46 E(X_0) \end{align}$$

Solve for $E(X_0).$

---

The answer can also be gotten by noticing that each time a $1$ occurs, the probability it is the last $1$ is $1/36.$

Answer 2

Since the time $T$ until we see the sequence of 1,2,3 is a stopping time for the sequence of die rolls, by Wald's equation we can just multiply $\mathbb E[T]$ by $\frac16$: the expected number of 1's from each die roll.