Doubling $100, priced: the chance of doubling before busting at every stake, every edge and every odds

The question

Most sessions have an unspoken target: double it, or go home. Put a number on it and the question becomes exact — with $100, a fixed stake and a known edge, what is the chance of reaching $200 before reaching $0? This is the gambler’s-ruin problem, and because our games publish their win chances and multipliers, it can be answered for each of them without a single assumption.

The stake is the lever

Take the simplest case: dice at a 50% target, ×1.98, a 1% edge. Bet the whole $100 once and you are at $198 or $0 with the chance of winning, 49.995% — as close to doubling as a ×1.98 bet can get in one step. Bet smaller, aiming for $200, and two things happen — the session takes more bets, and every bet hands over 1% of the stake on average. The edge is a toll per bet, and small stakes pay it more often.

Target $200 or more; bust when the bankroll can no longer cover the stake. On a fair coin paying ×2 every row would read 50%. The gap between 50% and each figure is what the 1% edge costs over the number of bets the session takes to resolve.

Read the last two rows twice. At $1 a bet, $100 doubles about one time in nine; at 50 cents, about one time in sixty. Nothing about the game changed between the rows — only how many times you let the house take its 1%. “Play small to make it last” is true: the session lasts 7,600 bets at $1. It just almost never ends at $200.

The edge compounds through the bets

A bigger edge does the same thing faster. Here are our even-money-style games at two stakes, with the chance of doubling $100.

Chinchiro’s payouts are not all even money (×3, ×4, ×6 for specials); it is modelled here as an even-money bet with the same return, which is slightly pessimistic for the long-shot payouts and does not change the picture.

At $10 a bet the games are a few points apart. At $1 a bet a 1% game gives you one chance in nine and a 2.8% game one in 240. The difference between a 1% and a 2.8% edge looks like 1.8 points on a payout table; over a session of a thousand bets it is the difference between a long shot and no shot. This is also why the question “which game has the lowest edge?” matters far more to a small-stakes player than to someone making one bet.

The odds change the answer

Everything above is even money. Dice lets you pick the odds at the same 1% edge, and that changes the doubling chance in a way most players would not guess.

Replayed with the engine’s multipliers; at $1 a bet the 1% target reaches $200 with two hits (the second one landing before 100 straight misses) and busts after 100 misses in a row.

At $1 a bet the coin flip doubles you 11% of the time and the 1-in-100 shot 41% of the time — nearly four times as often, at the same edge. The reason is the same toll: a long shot settles the question in far fewer bets, so the house collects its 1% far fewer times before you are at $200 or $0. At $10 a bet the order flips, because ten misses at ×100 is a bust before the first hit is likely.

What this is not

It is not a strategy. There is no stake, odds or game in these tables that reaches 50%, because the edge is paid on every bet and no sequence of bets avoids it — the martingale and its cousins only move the loss into a rare large one. What the tables do is make the trade visible: betting bigger buys a better chance of the target at the cost of a shorter session; betting a long shot buys the same thing at the cost of a wilder one.

The honest use of the numbers is to pick the session you actually want before it starts. A player who wants an hour of play at $1 should know that the hour is the product and $200 is not; a player who wants to double should know that one bet at 49.995% to $198 is the best price on the board and anything slower is paying for the privilege of waiting. Our risk of ruin piece covers the same trade in prose; this one puts the figures on it, game by game, from the engines that pay them.