You have a fair nine-sided die, where the faces are numbered from $1$ to $9.$ You roll the die repeatedly, and write the number consisting of all your rolls so far, until you get a multiple of $3.$ For example, you could roll an $8,$ then a $2,$ then a $5.$ You would stop at this point, because $825$ is divisible by $3$, but $8$ and $82$ are not.
Find the expected number of times that you roll the die.
I am fairly new to the concept of expected value, and I don't really know how to go about solving this. It would be great if someone could help.
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