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Should you toss a buck on longshots with high house advantage?

14 May 2012

Some slot machines and bonus side bets at table games have extremely high payoffs with correspondingly low probabilities of success. In this sense, the jackpots are longshots. However, the wagers aren’t simply all-or-nothing affairs. Rather, they generally offer ranges of payoffs with chances of success that rise as the amounts in question fall. Consequently, players can and frequently do earn dough on these gambles without having to be favored with extraordinary luck.

Twos and twelves at craps and single spots at roulette are more in the classic longshot tradition. They either win or lose. No consolation prizes are awarded for intermediate results.

The craps twos and twelves are 35-to-1 propositions that pay 30-to-1. The large offsets between odds of winning and payoffs give the casinos 13.9 percent edge, explaining why dedicated dice doyens denounce them as “sucker bets;” for reference, edge is under 0.5 percent on line bets with Odds at the same game. Straight-up roulette bets are 37-to-1 affairs at double-zero and 36-to-1 propositions on single-zero wheels, paying 35-to-1 in either case. The offsets are smaller than on the two or twelve at craps. But they’re still sizeable and lead to 5.26 and 2.70 percent edge at double-and single-zero tables, respectively, the same as for other wagers at these games.

Solid citizens frequently bet longshots like these despite the uphill battle against odds and edge. This, because the laws of probability tend to be overshadowed by the utility principle. The former would have decisions based on objective statistical trade-offs. The latter is a more subjective matter of individuals regarding a dollar – or whatever they risk on the bet – as chump change it won’t hurt to lose, while the reward is enough to be coveted even if it rarely materializes.

When series of bets are made on longshots with high values of house advantage, though, the laws of probability offer up results that may surprise many purists. The wagers may actually be more promising than the nay-sayers imagine.

Make believe a craps aficionado sets aside $30, and bets it $1 at a time on the two. The plan is to run through the $30 once, locking up and not re-betting any payoffs in the process. Chances are 42.9 percent, under half, of losing the whole shebang in 30 straight misses. They’re accordingly 57.1 percent, over half, of winning at least something. In particular, the outlooks for ending the exercise in the money are 36.8 percent of one win with a $1 profit, 15.2 percent of two wins with a $32 net, and 4.1 percent of three wins with a $63 gain. Greater earnings are also possible. Chances start slightly under 1 percent for four wins and a $94 payday, and plummet to the nano-percent realm for 30 wins and a $900 take. The theoretical price paid to the casino for the 30 $1 tries, in terms of the monetary equivalent of the edge for the action, is $4.17.

Alternately, pretend a roulette buff at a double-zero table reserves $35 and drops $1 at a time on a single spot. It doesn’t matter whether it’s the same or a different spot on successive spins. The chance is 39.3 percent, substantially under half, of never getting a hit and losing the $35. The complementary 60.7 percent is the prospect of finishing to the good. This figure includes 37.2 percent likelihood of winning once and earning $1, 17.1 percent of winning twice and gleaning $37, 5.1 percent of winning three times and making $73, and 1.1 percent of winning four times and pocketing $109. The remaining 0.212 percent is the probability of copping between $145 and $1,225. For this, players donate a theoretical $1.84 to the casino coffers in the form of edge.

Inquiring minds undoubtedly want to know the extent to which the lower edge at single- than double-zero roulette affects the outcomes when the strategy is applied to straight-up longshots. For 35 successive spins at $1 each, the chance of a total $35 wipe-out is 38.3 percent (compared to 39.3 percent); the likelihood of any net gain is the complementary 61.7 percent (compared to 60.7 percent). One win for $1 comes in at 37.3 percent (compared to 37.2 percent), two wins for $37 have a 17.6 percent (compared to 17.2 percent) chance, three wins for $73 have a 5.4 percent (compared to 5.1 percent) chance, and four wins for $109 have a 1.2 percent (compare to 1.1 percent) chance. The probability of grabbing between $145 and $1,225 is the remaining 0.238 percent (compared to 0.212 percent). The penalty imposed on players by the house advantage for the 35 bets is equivalent to $0.95. Lower edge helps, but hardly enough to notice in 35 spins.

So... sucker bets or auxiliary wagers worth a tumble? Decide for yourself, keeping mindful – of course – of this applaudable admonition from the admirable odemonger, Sumner A Ingmark:
Reluctancy to take a leap, when cost of doing so is cheap,
Sends those who will not go for it, away with naught to show for it.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.