For the same money at risk, should you concentrate or spread out your bets?

Doris and Horace enjoy shooting craps. They both start on the Pass line then make Place bets to cover additional numbers. Their Place strategies differ, however. Doris bets $10 on four or five, or $12 on six; Horace spreads $5 each over four and 10 or five and nine, or $6 each over six and eight. In terms of players’ theoretical loss from edge, the approaches are equivalent. Table 1 shows how much money this represents in 36 statistically-correct throws on the various choices.

                            pick up            drop      net loss to edge
Doris
$10 on four           3 x $18 = $54   6 x $10 = $60        $60 - $54 = $6
$10 on five           4 x $14 = $56   6 x $10 = $60        $60 - $56 = $4
$12 on six            5 x $14 = $70   6 x $12 = $72        $72 - $70 = $2

Horace
$5 @ on four & 10      6 x $9 = $54   6 x $10 = $60        $60 - $54 = $6
$5 @ on five & nine    8 x $7 = $56   6 x $10 = $60        $60 - $56 = $4
$6 @ on six & eight   10 x $7 = $70   6 x $12 = $72        $72 - $70 = $2

In something like 36,000,000 throws, the distribution of outcomes is apt to approach the statistically-correct proportions. This could also happen in a run of 36, 72, 108, or some other nominal multiple of 36 throws in a normal session – but it would be extremely rare. Enquiring minds might therefore want to know how the different strategies – less likely but larger payoffs as opposed to more likely but smaller returns, for the same expected frequency and size of misses – affect day to day results for individual solid citizens. Figures that seem relevant are a) the chance of achieving a particular win goal before busting out at a stated loss limit, and b) the probability of still being in action on a given bankroll after some specified number of rounds.

To find typical values, ignore wagers on the Pass line and imagine that Doris and Horace make single or paired Place bets totaling either $10 or $12 in the modes described. Pretend, further, they each start with $200 bankrolls – roughly 15 to 20 times the amount at risk on every throw. They’ll quit if they double their dough. Or, if they don’t attain the $200 gain level, they’d like to have at least three hours of thrills and chills – about 300 throws – for their $200 investments. Prospects of winning $200 before losing $200 and of remaining in action for at least three hours on a $200 bankroll are shown in Table 2 for the two modes of play on the indicated bets.

                   $10 on   $5 @ on       10 on    $5 @ on         $12 on  $6 @ on
                   four     four & 10     five     five & nine     six     six & eight
earn $200 before
losing $200        18%      10%           24%      17%             39%     36%

stay in action at
least three hours  84%      90%           89%      94%             87%     92%

Compare the data for the corresponding single and pairs of numbers (e.g., $10 on five and $5 each on five and nine). Grabbing more dough less often, the former case, gives Doris a leg-up over Horace with respect to reaching a win goal before butting into the loss limit. This reverses when the objective is to get more time at the rail on a given bankroll; Horace has a better shot than Doris at surviving the normal downswings of a session without exhausting his stake.

Table 2 also shows that going from outside to inside numbers – four to five to six, or four and 10 to five and nine to six and eight – improves a bettor’s chance of arriving at an earnings target before going broke, presumably because the combined effect of lower house advantage and higher frequency of successful coups outweighs that of smaller payoffs. The trend for session duration migrating from outside to inside is somewhat anomalous in that, for either betting scheme, prospects are more promising on fives or fives and nines than on fours or fours and 10s, but less so for sixes or sixes and eights than for fives or fives and nines. The latter situation is a consequence of the larger required bets on six or six and eight than on five or five and nine. To verify this, the wagers can be equalized at $30 to win $42 on the five and $30 to win $35 on the six. Here the probabilities of surviving for at least 300 throws are 36 percent on the five versus 43 percent on the six – increasing progressively from the outside to the inside.

This is a form of a classic dilemma for casino buffs. Whether to aim for a greater profit or a longer run on a finite entertainment budget. The Romans shrugged off this kind of quandary by saying “de gustibus non est disputandum” (there’s no disputing taste). The populist poet, Sumner A Ingmark, had a more mundane take: