When should we stake it all?

The Gambler’s Dilemma

A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

THE DILEMMA

When the stakes are high and time is not a luxury, finding a solution can be like gambling with fate. This was the scenario for Mike, needing £216 to settle an urgent debt, with only £108 in hand. The roulette wheel beckoned as a potential salvation, but what was the most effective strategy to double his money?

UNDERSTANDING THE ODDS IN ROULETTE

To fully grasp the situation that Mike finds himself in, it’s crucial to examine the mechanics and probabilities of the game he’s chosen as his lifeline: roulette. Specifically, we are considering a single-zero roulette wheel, a version of the game commonly found in European casinos.

Roulette consists of a spinning wheel and a small ball. The wheel is divided into 37 compartments or ‘slots’: numbers from 1 to 36 (randomly assigned as red or black) and a single zero slot. Bets can be placed on a single number, colour, or various combinations thereof.

In a single-zero roulette wheel, the player has a 1 in 37 chance of correctly predicting the outcome. This is because there are 37 slots in total: 36 numbers and the zero. So if you bet on a single number, the odds of the ball landing on that number are 1 in 37, or 36/1. The payout for such a bet, however, is 35/1. This discrepancy between the actual odds (36/1) and the payout odds (35/1) is where the house gains its edge. Every time a player wins, the house pays out less than the actual odds would dictate. In this way, the house earns a profit over time.

The ‘house edge’ is approximately 2.7%, a figure derived from the ratio of the single zero slot to the total number of slots (1/37). This constant advantage in favour of the casino is what makes the game fundamentally a game of negative expectation for players.

To understand the house edge in another way, consider this: if you were to place a £1 bet on each of the 37 slots, totalling £37, your return would be £36 (the £35 returned on the winning number plus the stake of £1). So for every £37 wagered, you would lose £1 using this strategy, which is approximately a 2.7% loss—exactly the house edge.

In conclusion, roulette, like all casino games, is a game of probabilities. And these probabilities, owing to the discrepancy between the actual odds and the payout odds, are slightly skewed in favour of the house. This fundamental understanding of the game’s odds is pivotal when contemplating betting strategies, as we will see with the employment of ‘bold’ and ‘timid’ approaches.

THE BOLD STRATEGY: STAKING IT ALL

Mike’s precarious situation leads him to contemplate a high-risk, high-reward approach known as the ‘bold’ strategy, which involves wagering all his available money at once. In this instance, he considers staking his entire £108 on the colour Red, a bet with almost a 50-50 chance, as the roulette wheel has 18 red slots out of 37 total slots.

To fully appreciate the audaciousness of this approach, it’s essential to understand the mathematics behind it. When betting on a colour, there’s a near-even split of potential outcomes: 18 red slots, 18 black slots, and the zero slot. Thus, the likelihood of the ball landing on a red slot is 18 out of 37, or roughly 48.6%. Consequently, with this single bet, he has about a 48.6% chance of doubling his money and obtaining the £216 he urgently needs.

However, it’s important to note that this is a single-round probability. Unlike a ‘timid’ strategy, where multiple rounds are played, the bold strategy is a one-off scenario. Therefore, the 48.6% chance of winning must be interpreted as his overall chance of achieving his target sum. There are no second chances or opportunities to recoup losses; it’s an all-or-nothing situation.

By putting all his money on one bet, he is maximising his return if that bet is successful. This is in contrast to a timid strategy, where the payout would be spread over multiple smaller bets, with the likelihood of achieving the target sum being significantly less.

But the bold strategy also comes with the highest level of risk. If the ball doesn’t land on Red, Mike loses everything. His entire available funds are at stake, making the potential loss just as significant as the potential gain.

In conclusion, the bold strategy is a high-stakes, high-reward approach. It encapsulates the old saying, ‘Who dares, wins’, and, in this case, provides him the best chance of reaching his £216 target. Why is this so?

TIMID APPROACH: MULTIPLE SMALL BETS

As opposed to the bold strategy, he could consider dividing his available £108 into 18 separate bets of £6 each. These small, successive bets would be placed on a single number until he either depletes his funds or hits the winning number, which would yield a payout of 35 to 1, giving him the £216 he needs.

To fully understand the implications of this strategy, we need to analyse it in detail. The probability of winning a single number bet in roulette is 1 in 37, as there are 36 numbers and one zero. Hence, for each individual bet, John has a 1 in 37 chance of winning, or approximately 2.7%.

However, the timid strategy involves making multiple small bets, and so we must calculate the probability of these successive bets all losing. Since each individual bet has a 36 in 37 chance of losing, the probability that all 18 bets lose would be calculated as (36/37) to the power of 18, which equates to around 0.61, or 61%.

As such, the probability of him winning at least once using this timid strategy would be equal to 1 minus the losing probability. Hence, the chance of hitting the target £216 is 1 − 0.61, or 39%.

Interestingly, the timid strategy, although appearing less risky, significantly reduces Mike’s chances of achieving his target sum compared to the bold approach. By spreading out his available funds across multiple bets, he lowers his exposure to loss in each individual game, but also decreases the likelihood of achieving his overall goal.

This strategy extends the length of play and the suspense, providing more instances of potential winning and losing. However, each bet also exposes Mike to the house edge, and therefore the risk of losses incrementally increases.

In this way, the timid approach offers more sustained engagement with the game but sacrifices the higher winning potential found in the bold approach.

THE POWER OF BOLD PLAY: TAKING A CALCULATED RISK

To look at it another way, consider a scenario where equal amounts are bet on red and black in each round. In most cases, the outcome will lead to breaking even, specifically 36 out of 37 times. However, when the ball lands on the single zero slot, the entire bank is lost. The more games played, the greater the chance of this happening.

By limiting the game to a single spin, the bold strategy minimises the number of times the house edge comes into play. Hence, playing fewer rounds decreases the likelihood of the house edge depleting the funds before reaching the target.

This strategy is not just about boldness in the face of risk, but more about understanding and working around the inherent disadvantage players face in casino games. By playing fewer games, you reduce the opportunities for the house edge to work against you.

CONCLUSION: THE INTUITION BEHIND BOLD PLAY

The intuition behind bold play in unfavourable games is grounded in a nuanced understanding of the mechanics of casino games and their built-in house edge. Bold play aims at striking hard and fast, capitalising on the relatively high chance of achieving the target sum in a single round, instead of facing the progressively increasing exposure associated with multiple rounds. In this sense, it’s a calculated and strategic form of boldness.