A Shuffle Through the Gaming Mailbag: Chuck-a-luck and Slot Payback Percentages

Q. I found a chuck-a-luck game online and have a question about how the odds change when betting two numbers. I started betting 3 and 4 together over and over. I slowly started increasing my bank. The payoffs on the game were: Single 1:1; Double 2:1; Triple 10:1.

I did manage to hit one triple, but most of my bank increase came from hitting two singles and doubles. I did not place any other bets, just 3 and 4 together. Naturally, if I hit only a 3 or 4, I would break even, but I made money whenever I got doubles or triples, or when I got singles of each number.

A. Playing two numbers at a time does not change the odds at chuck-a-luck. If it did, the casino would be an underdog anytime two players consistently played different numbers. That situation wouldn't last long.

For those unfamiliar with the game, it's also known as "bird cage." Three dice are placed in a cage. Players bet on numbers and the cage is turned over. The dice tumble to the bottom of the cage, and when they settle, bets are paid off. Chuck-a-luck is rare in Midwest casinos but is easily found online.

The pay table the reader found online, with one die of the wagered number paying even money, two paying 2-1 and all three on the same number paying 10-1, is good for chuck-a-luck but not great compared with other casino games. The house edge on single-number bets on chuck-a-luck with that pay table is 4.63 percent, and it stays at 4.63 percent regardless of whether you're betting one, two, three, four, five or all six numbers at a time.

There are 216 possible combinations of three dice -- 6 times 6 times 6. Of those 216 rolls, a single number will come up on one die 75 times, on two dice 15 times and on all three once. The bet will lose 125 of 216 times.

If you bet both the 3 and the 4, and see all 216 possible combinations come up in sequence, you will win once on three 3s, once on three 4s, 15 times on two 3s, 15 times on two 4s, 75 times on one 3 and 75 times on one 4, and in the long run the house will take 4.63 percent of your money.

However, we will have more frequent winning rolls when we bet two numbers at once than when we bet only one, because sometimes a roll that is a total loser on 4 will include a 3 -- or two, or three -- and sometimes a roll that is a total loser on 3 will include a 4, or two, or three. Instead of losing our total bet 125 of 216 times, we lose both bets only 58 of 216 times.

What that means is that extended winning streaks will happen more often than when we bet only one number. The winning streaks will tend to be less dramatic than a winning streak on a single number, because wins on one number are often at least partially offset by losses on the other. In a good streak, we will slowly increase our bankroll. However, when the inevitable cold streaks come, we will crash quickly because we're losing two bets at once.

Overall, betting two numbers at once will give us more frequent winning sessions than betting one number (with smaller, steadier wins), but also will give us bigger, faster losses in cold spells. In the long run, they balance out -- we lose 4.63 percent of the total amount wagered either way.

It's the same deal as with a roulette player who plays black and one of the dozens at the same time, or a craps player who covers all the place numbers. Wins will be more frequent as you cover more numbers, leading to more steady winning sessions, but losses will be steeper when they come.

Q. Would you happen to have any current information as to which casino in the Chicago area pays out the best percentage rates overall on the dollar machines, or any other info about where the slot players might have a slight edge from one casino to another? Is there a publication that tracks these numbers which is available to the general public?

A. The statistics you're looking for are released monthly by the Illinois Gaming Board and Indiana Gaming Commission and are posted on their Web sites. For Illinois casinos, go to www.igbstate.il.us and click on "Monthly Reports." For Indiana information, go to www.state.in.us /gaming and under "Reports," click on "Monthly Revenue."

The Illinois reports break down casino hold percentages for each denomination of slot machine - subtract the hold percentage from 100 and you get the payback percentage. The Indiana site does not list percentages but it does list coin in and casino win. Divide casino win by coin in and multiply by 100 to get the hold percentage, then subtract hold percentage from 100 to get payback percentage.

I'm overdue for a full column on slot payback percentages, so I'll do a breakdown next month. For now, let's look at dollar slots per your request.

In March in Illinois, Grand Victoria in Elgin led Chicago casinos, paying 94.96 percent on dollars. In northwest Indiana, Trump and Majestic Star in Gary were co-leaders at 94.1 percent.