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Why small wagers on high-edge side bets may not be as bad as some say

10 December 2012

Experienced casino buffs – solid citizens who are either ahead of the game or at least have enough moolah still stuffed in their mattresses to be able to afford repeat visits – know that side bets give the bosses an edge for which “usurious” is an understatement. So, maybe you wonder why so many folks – indeed, why you – make these wagers anyway.

The reason isn’t just the high potential payoffs. Nor is it merely a manifestation of utility theory, that essentially says remote probabilities aren’t a deterrent when you’re putting peanuts up for grabs on a shot at dough it’s doubtful you’ll ever earn otherwise. Both are factors, though. A more extensive explanation lies in what math mavens call the first three “moments” of the distribution. During the statistically few trials characterizing most casino sessions or visits, edge (first moment) is dominated by volatility (second moment) in outcomes with high positive skewness (third moment). In plain speak, when side bets are low relative to primary wagers, frequent losses won’t break players’ bankrolls but rare wins can yield substantial rewards.

The “Fire bet” you may find at craps affords an example and an object lesson on counter intuitive phenomena. Fire bets are wagers that shooters will make at least a designated number of different points before missing-out. Bets are paid for three or more different points in some casinos and four or more in others. Order is irrelevant. And the bet doesn’t lose if points are duplicated during a long hand, provided the requisite number of unique hits is also made. So, for instance, passes on five, eight, four, and nine count as four toward the payout, as does making five, eight, five, four, four, and nine.

The probabilities associated with Fire bets are inherent in the rules of craps. However, casinos may establish sets of payouts arbitrarily. The nearby table gives the probabilities a shooter will make zero to six unique points, along with three (A, B, and C) of the many ways payout schedules might be configured. The table also shows, for each set of payouts, the house’s edge and the standard deviation – the latter a measure of volatility you can picture as the representative bankroll jump per dollar bet per decision, irrespective of direction. Skewness can be seen qualitatively by the huge probabilities of $1 losses compared to the slim chances of large gains.

Alternate configurations of the Fire bet at craps.

Points   Probability   Payouts A   Payouts B   Payouts C
0            59.394%       -$1         -$1         -$1
1            26.075%       -$1         -$1         -$1
2            10.128%       -$1         -$1         -$1
3             3.343%       -$1         -$1          $5
4             0.880%       $25         $10         $25
5             0.164%      $250        $200        $200
6             0.016%    $1,000      $2,000        $500
edge                   -19.70%     -24.86%     -15.96%
standard deviation      $16.47      $26.78      $10.64

The tabulated data confirm that the house advantage is extremely adverse for these propositions. In contrast, for instance, the edge on Pass or Don’t Pass with triple Odds is under 0.5 percent. The casino earns a theoretical $0.14 per decision on a $10 line bet with $30 in Odds, but almost $0.25 on a $1 Fire bet (with Schedule B). Over half-again as much for $1 than for $40 at risk. Say you go to a $10 craps table with a $500 stake. You allocate, mentally or by actually separating the checks in your rack, $50 for $1 Fire bets and $450 for $10 line bets with $30 Odds and one or maybe two $10 or $12 Place bets. What are your chances of reaching or exceeding $1,000 profit on the Fire bet before exhausting the $50 you designated for this wager?

You’re not exactly sitting pretty no matter which of the representative payout schedules is in force. With “Payouts A,” your chance of success is 2.09 percent. With “Payouts B.” it’s 3.29 percent, and with “Payouts C,” it’s 0.827 percent. The prospects are low. But how often do you recall earning $1,000 making $10 line and place bets – or seeing anybody else do it, either? Further, since the Fire bet would be made when the dice move to a new shooter, the $50 set aside for it should last for a session of three-to-four hours even if every round is a $1 loss. You’re more apt to scrape the bottom of your $450 fanny pack before completing three or four hours making Line and Place bets at the $10 level than to lose your $50 on Fire bets.

A further factor that pops out of the data involves the interaction of edge and volatility with few enough coups that the latter swamps the former. The greatest chance of reaching the $1,000 mark occurs with “Payouts B,” which gives the house the highest edge but has the largest volatility. The slightest chance of achieving this win goal is with “Payouts C,” which gives the house the lowest edge but has the least volatility. The driving element is clearly volatility, not edge. All of which illustrates the insight intimated by the inkster, Sumner A Ingmark, that:

A good strategy intentional,
May flout wisdom that’s conventional.
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.