Doey-Don't is one name for a system or strategy sometimes used in the game of Craps - specifically Casino Craps, as found in the Casinos of Nevada or Atlantic City.
The idea behind the system is that by betting on both the Pass Line and Don't Pass Line (and also usually the Come and Don't Come bets) a passed or missed roll will be offset, resulting in a push. Profits are made by betting on the Pass, Don't Pass, Come, or Don't Come Odds, which pay True Odds instead of Even Money.
For the original bet, you place a bet of 1/2 unit on both the Pass and Don't Pass. Most casinos will label the Don't Pass as bar 12, which means that if a 12 is rolled on the come out then the Pass bet loses and the Don't Pass bet pushes, resulting in the loss of 1/2 unit. This is the only way any money is lost on the original wager.
By dividing the 1/2 unit loss by 36 (the number of possible rolls), you get -0.0138, which means the house edge when using this strategy is around 1.38% before taking the amount of odds wagered on into consideration. This gives the player a slim advantage over betting only the Pass or Don't Pass lines.
Most casinos offer 3x4x5x odds which means that if your 1/2 unit is $5 (common table minimum in Vegas) then if the point is 4 or 10 you can wager $15 in odds, on 5 or 9, the wager rises to $20, and on 6 or 8 you can place $25 in odds. Winning these bets pays $30 (2-1 on the 4/10, 3-2 on the 5/9, and 6-5 on the 6/8). Since these bets are paid at true odds, there is absolutely no house advantage.
Many gamblers will combine this system with a $1 bet on the 12. This changes the houses edge significantly to the house's benefit - you now have 35 chances of losing 1/10 of a unit, and one chance to win 2.5 units (the 12 bet pays 30-1, giving you 3 units, the pass bet loses .5 units). 2.5 - 3.5 = -1 unit, and the house edge more than doubles to -0.0278 or 2.78%. In other words, this is a sucker bet.
While the Doey-Don't strategy has only a slight reduction in the house edge over a standard Pass line bet, it offers a safer way to get to the odds bets, which carry zero house edge. Thus the slight reduction in the house's edge, if played at a table which allows for 10x or even 100x odds can result in the closest thing the casino will offer as a fair game. If you play this system at a table which allows 100x odds, then the house edge is reduced to about 0.014% - much lower than the house edge at six-deck blackjack when playing basic strategy.