Roman Numerals II

A random generator produces a sequence of symbols drawn from the set {I, V, X, L, C, D, M, #}. Each item in the sequence is determined by selecting one of these symbols at random, independently of the other items in the sequence. At each step, the seven letters are equally likely to be selected, with probability 14% each, but the # symbol only has a 2% chance of selection.
We write down the sequence of letters from left to right as they are generated, and we stop at the first occurrence of the # symbol (without writing it). However, we stipulate that what we have written down must always (when non-empty) be a valid Roman numeral representation in minimal form. If appending the next letter would contravene this then we simply skip it and try again with the next symbol generated.
Please take careful note of About… Roman Numerals for the definitive rules for this problem on what constitutes a “valid Roman numeral representation” and “minimal form”. For example, the (only) sequence that represents 49 is XLIX. The subtractive combination IL is invalid because of rule (ii), while XXXXIX is valid but not minimal. The rules do not place any restriction on the number of occurrences of M, so all positive integers have a valid representation. These are the same rules as were used in Problem 89, and members are invited to solve that problem first.
Find the expected value of the number represented by what we have written down when we stop. (If nothing is written down then count that as zero.) Give your answer rounded to 8 places after the decimal point.

The formulation of this problem is very complex as it combines knowledge of probability and randomness theory with Roman numeral rules. For us to produce correct and accurate result, we need to consider how the probability of generating valid Roman numerals combines with the chance that the random generator will create a ‘#’, stopping the generation of numbers.

Given the constraints of the problem, it’s not possible to simply calculate the solution with a standard formula or mathematical operation. This problem would actually require creating a smart simulation or establishing a set of interconnected probability equations factoring in all possible sequences of Roman numerals and their corresponding values.

Moreover, given the complexity of Roman numeral rules (i.e. subtractive combinations like IV instead of IIII), the problem becomes exponentially complex as the size of the number increases, particularly since the # could appear at any point.

To tackle this problem, one potential course of action would be to code a program that simulates this random generation. You could then run the simulation repeatedly (say, several million times) and then calculate the average value from that data.

For a more precise result, it’s probably necessary to set up complex, cascading probabilities taking into account sequences of numbers and then summing up these all probabilities multiplied by their representative numerical values. This would likely need to be calculated programatically, potentially using dynamic programming techniques, and would remain a significant challenge.

Without explicit calculations, it’s unfortunately unfeasible to simply provide a single numerical answer. I suggest seeking assistance from someone proficient in mathematical programming for this kind of advanced combinatorics problems if you’re seeking the exact result.

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