Is there an analytical solution to this probabilistic problem?

I am a big fan of MMORPGs, and this one, Cabal Online, has a particularly punishing upgrade system. Items can be upgraded to increase their stats, ranging from level 0 to level 15 (for simplicity). Upgrading costs cores (can be interpreted as currency). Attempting an upgrade can result can result in the item’s level going … Read more

Inconsistency when applying the Kelly Criterion

Context So I’m doing some research regarding the Kelly Criterion. By considering a coin tossing game in which you have even money odds where there is a probability p > 1/2 of winning, a probability q < 1/2 of losing and you bet the fraction \ell of your capital in each bet. Then, your capital … Read more

Is Keno a fair game?

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with probability, which perhaps yields the shortest, simplest proofs, but other than that, the textbook gave no hints really and I’m really not sure about how to approach it. Any … Read more

5×55 \times 5 grid with coins and bombs. What is the optimal gambling strategy?

I’ll start off by saying that this is probably not deemed an adequately advanced question for this site, and I’ll probably phrase this poorly, I apologise if I do. Suppose there is a 5×5 grid [∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗] On this grid, coins will be placed on some of the squares, and bombs will be placed on the … Read more

Why will repeated betting on a fair coin toss lose you money in the most cases? And how can you calculate it?

Imagine you play a game where you can set a certain amount of money, and you have a 50% chance of doubling your bet and 50 % of losing your bet. (To make the game interesting to play you can also give a chance of winning higher than 50%). You can play the game multiple … Read more

Gambler’s Ruin variant: each bet is for 1/k dollars, what happens to probability of winning as k approaches infinity?

I am trying to a solve a variant on the Gambler’s Ruin problem, in which two gamblers A and B make a series of bets until one of the gamblers goes bankrupt. A starts out with i dollars, B with N−i dollars. The probability of A winning a bet is given by p, with 0<p<1. … Read more

Determining the statistical significance of the performance of a gambler

Imagine someone claims they win significantly more than they lose when betting on roulette. Presuming that it were possible to have a winning system how could you calculate the statistical significance of their claim? The roulette table has no “edge” (i.e. no zero/double zero…). The issue comes with the fact that they are able to … Read more

bounded gambling systems (Theorem 4.2.8 in Durrett: Probability Theory and Examples)

Theorem 4.2.8 in Durrett: Probability Theory and Examples states Let Xn,n≥0 be a supermartingale. If Hn≥0, is predictable and each Hn is bounded then (H⋅X)n:=∑nm=1Hm(Xm−Xm−1) is a supermartingale. While I know an example showing that this statement does not hold when Hn is not bounded, I cannot see where we use boundedness in the proof. … Read more