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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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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 $E(X) = 1/p = 3$

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  • $\begingroup$ If I remember correctly $E[X]=\frac{1-p}p$ $\endgroup$ – Shubham Johri Dec 10 '19 at 22:25
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    $\begingroup$ To add on to your answer, whenever we don't get a number divisible by $3$, we arrive at a number that is equivalent to $1$ or $2$ modulo $3$. In the first case, we need to roll a number that is equivalent to $2$ modulo $3$; in the latter, we need a number equivalent to $1$ modulo $3$. In both situations we have a $1/3$ chance to create a number that is divisible by $3$, which is why it is independent of the value we are currently at. (+1) $\endgroup$ – WaveX Dec 10 '19 at 22:27
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    $\begingroup$ @ShubhamJohri I believe the distinction lies in whether we count the last dice roll. If we do it's 3; if not it's 2. $\endgroup$ – marcelgoh Dec 10 '19 at 22:34
  • $\begingroup$ Please do not discuss this problem! This is an active homework problem. @Rose Callihan: I realize that homework may be challenging. If you wish to receive some help from the staff or other students, I encourage you to use the resources that the online classes provide, such as the Message Board. Thanks. $\endgroup$ – wonderman Dec 12 '19 at 17:23

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