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.
A number is divisible by 3 if and only if the sum of its digits are divisible by 3. Also, note that rolling a die with nine sides, we always have a 1 in 3 chance of getting a number that makes the sum divisible by 3 (because of the 9 values the sum could take, 3 must be divisible by 3). This is independent of the value we're currently at, so this is just a geometric random variable with $p=1/3$ and the expected number of trials is $E(X) = 1/p = 3$