Do your chances of winning get better if you keep playing?

Casinos have an edge or advantage over players because bets are structured with offsets between payoffs and odds that must be overcome to win. With $10 on the nine at craps, for instance, solid citizens are paid $14 when the dice show a nine and lose $10 when a seven pops. Other outcomes get no action. Nines are formed in four ways (3-6, 4-5, 5-4, 6-3) and sevens in six ways (1-6, 2-5, 3-4, 4-3, 5-2, 6-1). Odds against joy are therefore 6-to-4 (1.5-to-1). Payoffs are 14-to-10 (1.4-to-1). On the average, for every 10 win-or-lose decisions on $10 wagers, players’ fortunes rise $14 x 4 = $56 and fall $10 x 6 = $60. The $4 difference is what the casino earns from the edge. For craps bets on nine, that edge is 4 percent of the $100 gross bet on the 10 coups – equal to $4.

This doesn’t happen on every particular run of 10 decisions. The principle works for the bosses because they book so many wagers that, according to the laws of probability, frequencies of outcomes approach the theoretical values. After, say, 10 million decisions, sevens and nines will have occurred roughly six and four million times, respectively. Were these the precise counts, $10 bettors would have raked in $56 million and given up $60 million, leaving the casino with $4 million of the $100 million put at risk. Even after 10 million coups, though, the take won’t normally be exactly $4 million. But the deviation will be by only a small fraction.

Players, in contrast, may succeed despite the house advantage or lose far more than the edge would suggest. This, because they undergo few enough trials that actual frequencies of outcomes may depart markedly from the theoretical values. After 10 decisions on the nine, for example, six losses and four wins is the most likely count. But it isn’t overwhelmingly more so than other tallies. The accompanying table gives the outlooks for all possible sets of nines and sevens, along with the associated net player gains or losses, for 10 decisions with $10 bets.

# of nines	# of sevens	probability	net
0               10              0.60%         -$100
1               9               4.03%          -$76
2               8               12.09%         -$52
3               7               21.50%         -$28
4               6               25.08%          -$4
5               5               20.07%          $20
6               4               11.15%          $44
7               3               4.25%           $68
8               2               1.06%           $92
9               1               0.16%          $116
10              0               0.01%          $140

The “expected” four nines and six sevens, losing $4 on x $10 = $100 up for grabs, is the most likely upshot with 25.08 percent probability. The chance of five nines and five sevens, a $20 profit, isn’t far off at 20.07 percent. And the prospect for three nines and seven sevens, which will cost players $28, is similarly close at 21.5 percent. These figures indicate that the likelihood of departing from the theoretical amounts, up or down, is reasonably strong in the short run.

The bad news is that the probabilities in the table highlight an inherent bias, even with relatively little play. Craps buffs have 39.69 percent promise of profit on the nine by doing better than four wins and six losses. But they have the complementary 63.31 percent chance of losing. This includes the 25.08 percent chance of obtaining the theoretical numbers of wins and losses for a net $4 deficit, and the additional 38.23 percent threat of faring worse.

There’s good news, too. It’s true that, given sufficient action, casinos can be confident of a profit approaching the gross wager multiplied by the edge. However, subject to a critical caveat, few individuals will play enough to reach a point where their cause is hopeless. As illustrations, after 100 decisions, a player’s profit potential with Place bets on the nine drops only to 37.75 percent; with 1,000 coups, a player’s chance of triumph is still 14.35 percent – down but not out.

Oh yes, about that caveat. Bankroll limitations are a sword of Damocles. Of course, dice doyens aren’t apt to bet exclusively on the nine. But, assume arguendo that some do – betting $10 with $200 stakes. If a person could complete 100 rounds without going broke in the process, the chance of showing a profit is, as noted, 37.75 percent. But the player has over 15 percent chance of exhausting such a bankroll before completing this many rounds and accordingly being unable to recover. Likewise, if a person could complete 1,000 rounds, the probability of finishing with a profit is 14.35 percent, but this ignores a greater than 87 percent chance of running out of dough before bouncing back. Such premature busts increase the overall probabilities of losses. All of which contributes to the dilemma that the more you play, the more you might earn, but the probability keeps shrinking. As the poet, Sumner A Ingmark observed: