How Trying for Less Can Earn You More

28 June 2006

By Alan Krigman

Arnie and Marnie, apocryphal roulette enthusiasts, forever argue strategies for the game. One such debate involved taking $100 and betting $10 per spin, straight-up, on single numbers they got from their 1-900-PSYCHIC advisors. But, betting it differently.

Arnie insists it's best to put the $10 on the number and play until he either wins, at which point he'll quit, or loses the whole $100. Marnie's positive it's preferable to put the $10 on a spot and keep it there for 10 spins, regardless of what happens.

How do these systems compare?

At worst, each stands to lose $100. And they have the same chance of doing so. The probability this will be the case--that of 10 misses in a row--is 37/38 multiplied by itself 10 times, or 76.59 percent. This is slightly over three out of four times.

On the happier side, Arnie's methodology will produce a maximum payoff of $350 if he wins on the first spin. His earnings drop by $10 for each additional spin the wheel needs before finding his lucky number. The net reaches a minimum of $260 on the 10th try. The probability of winning anything at all is the complement of that associated with busting out, 23.41 percent. Prospects of success are brightest on the first coup, at 2.63 percent; they fall gradually and are 2.07 percent on the 10th spin.

For Marnie, a single win yields a $260 payday and has a fairly favorable 20.7 percent chance of occurring. That marginally exceeds one out of five. Wins on two spins, netting $620, have a far lesser 2.52 percent probability. Fatter profits are possible, all the way up to $3,500 for 10 successive wins, but with rapidly falling likelihood. Grabbing $980, for instance, comes in at 0.18 percent -- somewhat under two out of 1,000. A $3,500 show stopper is at about one out of 6 quadrillion. Marnie's chance of arriving home richer than she left, like Arnie's, is 23.41 percent.

Notwithstanding her potential for major moolah, Marnie's average revenue is a bit less than Arnie's. It's $71.33 versus $71.91. Intuitively, this is because her plan offers a relatively strong promise of winning only $260, with chances dropping precipitously as the amounts escalate to $3,500; Arnie, on the other hand, has a narrow range of chances for gains from $350 to $260.

Some solid citizens assume that quitting after a win cuts house edge in the game by depriving the bosses of the opportunity to get their money back. It turns out that the casino's expected income from edge is, indeed, lower for Arnie than Marnie $4.68 rather than $5.26 on the total of $100 they bring to the table.

Think of this as follows. Both have 76.59 percent chance of losing $100 in 10 spins. So the average on the debit side is $76.59. Arnie's average win was $71.91, giving the house the $4.68 difference as its theoretical profit. Marnie's average gain was $71.33, so subtracting loss and win values leaves $5.26.

The offset in rake doesn't arise from a reduction in house advantage, however. Edge is 5.26 percent either way. Marnie wagers a total of $100, $10 per spin for 10 spins. And 5.26 percent of $100 is $5.26. In contrast, by quitting after a win, Arnie doesn't always go for 10 spins. He averages only 8.895 spins and accordingly puts less money at risk in the long run. His handle averages $88.95 per game. And 5.26 percent of $88.95 is $4.68. The savings is therefore a consequence of betting less over the 10-spin sequence rather than of a change in edge.

Which alternative would the gurus say is better? It's your dough, and you now have the facts needed to weigh the options, so you're as good a guru as anyone in judging which is superior for you. Both schemes offer the same chance to win at least something and conversely, equal probability of losing $100. Arnie averages higher proceeds. Marnie has a much greater shot at a lower win and a hope, faint as it may be, of a major payday. A riddle reflected in this rumination by the rhymester, Sumner A Ingmark:

When deciding amongst variations,
Weigh both benefits and limitations.