Let's define a monopoly throw of k dice up to n as throwing k dice repeatedly until the sum is equal or superior to n. For example, a monopoly throw of 2 dice up to 15 could be as follows :
1st dice throw : 4 and 3, the sum is 7. 2nd dice throw : 5 and 6, the sum is 18, we're stopping there. Thus this monopoly throw reached 18.
I'm calling it a "monopoly throw" because it models moving your pawn on a monopoly board and hoping you get to a certain cell. If you pass the cell, too bad, but if you stop before, you can still hope to get to it in your next throw. I have simulated this (in a more general way) as well as a way to compute probabilities in the following Python program : https://replit.com/@Givrally/Monopoly-Throws
I have a few questions about this procedure :
Is there an actual name for this apart from "monopoly throws" ? It's a bit hard to get information of it when your google searches look like "probability of getting to the exact number when you're throwing a pair of dice until you get to that number of higher".
Is there a closed form for the probability of the monopoly throw stopping at exactly n ? There probably isn't, but it's worth asking.
If not, I have used the following formula to try and compute that probability : $$P(M=n) = \sum_{k=2}^{12}T_k*P(M=n-k)$$
with $T_k$ being the probability of getting a sum of k when throwing two dice. $T_k = \frac{6 - |7-k|}{36}$ when k is between 2 and 12, and 0 otherwise. My intuition is that it's the probability of getting to n-2 multiplied by the probability of getting a 2 this time, plus the probability of getting to n-3 multiplied by the probability of getting a 3 this time, and so on. Is this correct ? Or am I missing something ?
- Why does this probability seem to approach 1/7 as n grows ? I assume it's got something to do with the fact that the expected value for throwing two dice is 7, but I can't quite make the connection between that and the monopoly throw.