The change in value of the investment each year is modeled as follows:
Divided by 2: 1/4
Remain unchanged: 3/8
Doubles: 1/4
Quadruples: 1/8
Where I'm at: I'm aware that this needs to be formulated into a normal distribution, but I'm not sure how to come up with the right formula, or how to correctly approximate it to normal. Would really appreciate some insights!
No normal (even continuous) distribution would fit your evolution rules, which are clearly discrete (if only because $P(X_{t+1} = X_t) = 0 \not= 3/8$.
With your evolution probabilities, the best you can do IMO is to enumerate all possible values of the stock in year $n$ for $n \le 10$ and then compute the probabilities each year using a probability tree (this is Google's first result).
PS: you may want to enumerate values for $\log_2 (X_t/X_0)$ (if $X_t$ is your value in year $t$) so transitions from $t$ to $t+1$ are $-1$, $+0$, $+1$ and $+2$.
PPS: if really you wanted to fit a normal distribution, I'd suggest you compute the expectation and variance of your distribution for $X_t/X_{t-1}$ and use these as $\mu$ and $\sigma^2$ parameters for your normal dist. (with common notations, found for instance here).
But again, that would be a very poor fit (notice that your distribution is skewed, i.e. not symmetrical).
edit
To get the exact result (modulo numerical/rounding errors) with Excel (or any spreadsheet editor), here is a hint https://i.stack.imgur.com/vjSBT.png (look at I22's formula). Basically, noting $Y_t = \log_2 (X_t/X_{t-1})$, you go from one column/date $t$ to the next by using the fact that $$\displaystyle P(Y_t=a) = \sum_{i \in \{-1,0,1,2\}} P(Y_{t-1}=a-i)P(a-i \rightarrow a)$$ where the transition probabilities $P(a\rightarrow b)$ do not depend on $t$ ($Y$ is a Markov chain, but you don't need to know that to understand this simple case). Then copy-paste your formula over all possible values for $t = 1 \ldots 11$ and sum all probabilities in the last column corresponding to cases where $Y_{11} \le \log_2(30)$. Takes 5 minutes all-in.