Charles Hoffman | Searcharxiv

arXiv2015

Taking the risk out of RISK: Conquer Odds in the Board Game RISK

Dice odds in the board game RISK were first investigated by Tan, fixed by Osbourne, and extended by Blatt. We generalized dice odds further, varying the number of sides and the number of dice used in a single battle. We show that the attacker needs two more than 86% of the defending armies to have an over 50% chance to conquer an enemy territory. By normal approximation, we show that the conquer odds transition rapidly from low chance to high chance of conquering around the 86%+2 threshold.

math.HO↗

arXiv2015

Enumeration of k-Exceedance Lattice Paths with an Application to Comparing Chains of Order Statistics

We enumerate the number of monotonic lattice paths starting at $(0,0)$ and terminating at $(m,n)$ in which $l$ of the first $k$ steps lie below the line $y=x\ (0\leq k\leq m\leq n)$. These closed formulas consist of terms which are a product Catalan numbers, ballot numbers and binomial coefficients. We then apply the combinatorial formulas to failure analysis by deriving a probability distribution that compares the performance of a $k$-out-of-$m$ system to a $k$-out-of-$n$ system of continuous, independent, and identically distributed random variables. Lastly, we provide asymptotics in a few special cases of $k,m,n$ and leave others as conjecture.

math.CO↗