Important Pre-Requisite Knowledge
On the image board 4chan, every time you post your post gets a 9 digit post ID. An example of this post ID would be $586794945$. A Quad is a post ID which ends with 4 consecutive identical numbers. For example 586794444 and 586796666 is a Quad. A trip is a post ID which ends with 3 identical numbers. For example 586794333 or 586794555 are both trips.
My Question
a) What is the probability of receiving a trips post ID
b) What is the probability of receiving a quads post ID
c) How many posts are necessary (assuming each new post receives a new ID) for 3 trips to show up
These are questions I came up with while on the site and I'm looking to see if my answer is correct. I'm pretty sure I know part a,b. I'm having difficulty with part c though, looking for a way to solve that.
My Work
Part A
Our sample space is all possible posting IDs. Therefore $|S| = 10^{9}$
To calculate our $|E|$ we need to know all possible trips. We first pick our three ending letters (10 ways to do this). Then the $4^{th}$ to last digit must be different from the last 3 so we select it in 9 ways. We then have $10^5$ ways to select the starting 5 digits. Therefore, we have $10*9*10^5$ ways to select trips. Therefore, the probability of selecting a trips is $\frac{10*9*10^5}{10^{9}}$ = .009
Part B
Similar process to part A. We have the same sample space. 10 ways to select the quads. 9 ways to select the $5^{th}$ to last digit, and finally $10^4$ ways to select the remaining 4 digits. Therefore probability of $\frac{10*9*10^4}{10^9}$ = .0009
Part C
Don't really know where to begin. I'm thinking maybe there are $10*9*10^5$ possible trips and $10^9$ total IDs so maybe we have to post $10*9*10^5 – 10^9$ to ensure we get a trips.
Best Answer
a, b)
Be $X$ a random variable that counts the repeating numbers from the right. Take into account that when a post is a Quads is not a Trips.
$\begin{align} P(X=3)&=1\times\frac{1}{10}\times\frac{1}{10}\times\frac{9}{10}\\ &=0.9\%\\ P(X=4)&=1\times\frac{1}{10}\times\frac{1}{10}\times\frac{1}{10}\times\frac{9}{10}\\ &=0.09\% \end{align}$
That means, you choose the rightests number, then have $\frac{1}{10}$ the next one is the same, and so on, until the fourth or the fifth, which you want it to be different.
c)
Take into account the following. If you have a Trips ($xxxxxx111$), the next one will come when you sum $111$, being $xxxxxx222$. There are two exceptions, though. First is when you get through the Quads. Where you'll have to sum $222$ from one to the next one, since one doesn't count. The other is when you are in $xxxxxx999$, where you get the next Trips on your next post, $xxxxxy000$.
Anyway, the mean for that is easier calculated using the probability we just found. If we'd select random number of posts $t$, by mean we'd find $tP(X=3)$ Trips. Let's find for what number of posts, the expected number of Trips is $1$.
$\begin{align} 1&=tP(X=3)\\ &=0.0009t\\ t&=111.1111 \end{align}$
By mean, every $111.1111$ posts you'll have a Trips. For Quads it's equivalent.
$\begin{align} 1&=t'P(X=4)\\ &=0.00009t'\\ t&=1111.111 \end{align}$
By mean, every $1111.111$ posts you'll have a Quads.