The Parity Line Bet System is popularly called the 'Doey-Don't' after the label given to it in Frank Scoblete's classic book: Beat the Craps Out of the Casinos and Play Craps and Win, p. 39 (Bonus Books, 1991). To understand the Parity Line Bet System, you must first have a solid understanding of the Line Bets and of placing Free Odds. If you don't understand those bets thoroughly, what we're about to discuss will probably just give you a big headache.
You will recall that the Odds bets are great bets, and would be perfect bets except that you also have to make a Pass, Come, Don't Pass, or Don't Come bet before you can take advantage of the Odds. The Parity Line Bet System is a simple twist on the basic strategy which seeks to minimize the Pass, Come, Don't Pass, or Don't Come bet, by the simple expedient of betting equal, offsetting amounts on both the Pass and Don't Pass lines, or the Come and Don't Come bets. The idea here is that you essentially 'wash out' the Pass or Don't Pass bet, by making it's counterpart. Then, by placing as much Odds as is possible, you essentially water down the House's Advantage even further.
Craps Strategy For Beginners
Example: Same table with 3x Odds. Instead of making a single $10 Pass, you make a $10 Pass bet AND a $10 Don't Pass bet. If a 7 or 11 is rolled, your Pass bet wins but your Don't Pass bet loses, so that is a wash. if you roll 3 or a 12 (or a 2 in some casinos) then your Don't Pass bet wins and your Pass bet losses, so that washes out. If you roll a 2 (or a 12 in some casinos), then ONLY your Pass bet loses (your Don't Pass bet doesn't win). If 'numbers' (4, 5, 6, 8, 9, or 10) are rolled, you then place $30 Odds on either Pass or Don't Pass (whichever you choose). With the Parity Line Bet System, you odds of losing your Pass Bet are 1 in 36, which gives the House a 2.8% advantage. Note that this is HIGHER than the House Advantage for either the Pass/Come or Don't Pass/Don't Come Bets. The important difference is that those bets become bad bets on a little more than half the rolls, whereas with the Parity Line Bet System, it is only a bad bet the 1 in 36 times that a 12 shows up on the dice.
Keep in Mind: Even with the Parity Line Bet System, the dice are either bouncing your way or they are not. If the dice are bouncing your way, you would be better off betting the Pass/Come or Don't Pass/Don't Come bets to maximize your income. If the dice aren't bouncing your way, you won't lose as much, but you will still be losing. The Parity Line Bet System very slightly helps your chances over the basic game, but it doesn't automatically make you a winner, or even switch the advantage to your favor.
Analysis: One way to think of this is as a 'tax' which you pay every 36 rolls to allow you to play the casino with no advantage. But another way to look it this as a 36 roll 'Risk Cycle', i.e., every 36 rolls the Casino will be up on you one bet. Then, compare it to the Risk Cycles for the Pass/Don't Pass and Come/Don't Come Bets -- those Risk Cycles are nearly twice as long at approximately 70 rolls before the House can expect an advantage! So, you are really half as well off with the Parity Line Bet System as you are with the Basic Strategy (so there is no point in pursuing this strategy).
The Parity Hedge System