Is there a way to determine how much of a "fighting chance" I had?

Question

As a child, I was playing Monopoly in a local tournament against "Chuck" (the hero of some of my other game questions). Early on, he got a Monopoly of his namesake maroons: St. Charles Place, State Street, and Virginia Ave. After he acquired "stops" on all the other Monopolies, the game should have ended there except for one thing.

Toward the end of the game, a bystander, a girl named Martha, urged Chuck to give me a "fighting chance." He agreed to give me his defense to the Purple monopoly (Baltic and Mediterranean), in exchange for my defense to the orange Monopoly (St. James, Tennessee, New York). The other matters of note were that Chuck had hotels on the maroons, I had three railroads and one utility, and we each had about $1000 of cash.

Through what quantitative analysis (Monte Carlo simulation, perhaps), can I determine how much of a "fighting chance" I had at this point? I would guess that I would win less than one game in a hundred, perhaps less than one in a thousand, but in a million trials, I should have one or more wins.

Answer 1

A fat chance/A slim chance, whatever you wish to call it.

Even if you could get the last railroad AND the last utility AND get all hotels on your set for $0, you will still lose, unless a miracle happens, i.e. you never land on his hotels or on the set until his hotels are down.

There is a 7.54% chance (2 decimal places) you will land on his hotels (and pay at least $750) = equivalent to you losing $56.55 (2 decimal places) every turn (assuming you will land on his hotels at EXACTLY 7.54%)

There is a 4.29% chance he will land on your hotels (250 for Medit, 450 for Baltic) ($15.06/turn)

If you have the 4 railroads, 11.38% chance (gives $200) ($22.76/turn)

2 utilities: Average rent 7*10=$70, 5.41% chance = ($3.79/turn)

Total "gain": $41.61/turn

Total "loss": $56.55/turn

So this means that you may lose.

Note again that this is the most general case and I am only working with probabilities, that is, if the dice never or always leads to a space, this answer might be wrong.

Credits http://www.tkcs-collins.com/truman/monopoly/monopoly.shtml for all the probabilities