How Bet Size Affects Profit and Loss Potential

The relationship is obscured by low-probability high-value payouts like progressive jackpots on the slots and side-bets in Caribbean Stud. With these, the haul of everyone's hope hangs on the odds of winning as well as the amounts bet, and sowing single small wagers can reap rich rewards on one spin or deal.

The relationship is also cloudy in games where most wagers pay close to even money - like blackjack, craps, roulette, and baccarat. For instance, seasoned players know they should size their bets to their bankrolls, regardless of win goals and loss limits; overbet and they may get KO'd during normal downswings, underbet and they may not be happy with earnings during reasonable upswings. Further, solid citizens rarely keep bets constant in the heat of the action. And, since specific sessions are short compared with spans over which the laws of probability reliably predict results, individual wins or losses seem to defy logic.

To picture how bet size affects likely profits and losses, consider some extreme examples of a single betting situation, isolated from all extraneous factors. I'll illustrate with three alternate place bets on the nine at craps. Other propositions and more moderate amounts would yield variations on the same theme.

Say that Macey, Gracie, and Stacy each buy-in at a craps table with $100,000. They'll wager the money once-through on the nine, replacing bets from the original stakes after every decision - win or lose. Macey will make a hundred bets of $1,000 each. Gracie will place the nine a thousand times for $100 each. And Stacy will grind out ten thousand $10 nines.

What are the intrepid trio's prospects when the dust settles?

Each has 50-50 chance of being over or under $4,000 behind. The $4,000 loss is an average, representing the four percent edge the house earns when $100,000 gross is bet on the nine. Apportionment of the $100,000 during the session is irrelevant in this respect.

The difference between the three betting strategies lies in the chances of being various amounts better or worse than the nominal $4,000 damage. Results, here, are determined not by edge, but by the variance or volatility associated with every wager. The sizes and numbers of bets made during a session now become significant. For the same totals placed at risk over a period of time, fewer but larger bets increase fluctuations, raising the likelihood of finishing further above or below the average.

Macey, after a hundred heady tries, has a healthy 36.69 percent shot at being at or above break-even by ending with at least $4,000 more than the average. Gracie, after a thousand daring decisions, has a decent 14.12 percent chance of being ahead. Stacy, after ten thousand prudent place bets, has piddling 0.03 percent probability of earning a profit.

There's no hocus-pocus in this. The downside mirrors these figures. Macey runs a 36.69 percent risk of dropping $4,000 below the average - $8,000 or more in the hole. Gracie has a 14.12 percent likelihood of losing at least this much. But Stacy's chance of being below $8,000 is only 0.03 percent.

What conclusions can you draw from this example to help meet your individual gambling goals? Long-term, in casino games with negative expectations, the house edge predominates. The longer you play, the more it's the total you place at risk that matters, regardless of how you allocate your bankroll. So, many modest bets are apt to put you within a narrow range above or below the average. Short-term, however, volatility is controlling. A few big bets can send you into deep clover... or deep trouble. Envisioning just such multi-dimensional trade-off opportunities, Sumner A Ingmark, the Keats of compromise, wrote:

Facing many choices,
Leads gamblers to frustration.
But the heart rejoices,
On reconciliation.