A four-card royal is one of the best starting hands

Jim is a Colorado reader who e-mails me several times a year, usually with video poker conundrums.

His latest came after a session at 9/6 Double Double Bonus Poker, the full-pay version where full houses pay 9-for-1, flushes 6-for-1 and where the return with expert play is 98.98%.

"Three times, I was dealt four parts of a royal flush," he wrote. "I didn't get a single winner in the bunch. Not even a high pair. That got me to wondering. How good a start is four parts of a royal, anyway? There's no guaranteed winner. I'm living proof of that."

There are no guarantees, but four parts of a royal is one of the best starting hands around. The expected value differs depending on just what four parts you have. Let's say your fifth card is an unsuited 7. Then the average return is 98.40 coins per five wagered when you have suited 10-J-Q-K, 92.77 on suited J-Q-K-A or 92.45 on 10-Q-K-A, 10-J-Q-A or 10-J-K-A.

The expected value is highest on 10-J-Q-K because it offers an extra straight flush possibility, as well as additional possible straights. When your four-card royal start includes an ace, the only possible straights open to you are ace high. When you start with 10-J-Q-K, you can draw either ace-high or king-high straights.

With EVs exceeding 90 coins per five wagered on any of those combinations, you're in a stronger position with a four-card royal start than with any starting hand except a dealt royal flush, straight flush or four-of-a-kind. From full houses on down, no dealt winner is as valuable as starting with four cards to a royal flush. A dealt full house will return 45 coins for your five coin wager -- less than half the expected value with any four-card royal start.

However, Jim's experience is not unusual. Roughly half the outcomes with a one-card draw to a royal will result in no payoff at all. Let's say you start with the best possible four-card royal, 10 through king, with an off-suit card of 10 or lower as the fifth card. Your initial deal gives you five cards that no longer can be drawn, reducing the remaining deck from 52 cards to the 47 available for your draw. That leaves these 47 possible outcomes:

• One is the suited ace, giving you a royal flush and a 4,000-coin jackpot.
• One is the suited 9, for a straight flush and a nice 250-coin pay.
• Three are the off-suit aces, and three more are the off-suit 9s, for six possible straights, each worth a 20-coin return.
• Seven are low cards in your suit that give you a flush, but not a straight flush or royal flush. Each flush brings you 30 coins.
• Three are jacks, three are queens and three are kings, each giving you a high pair that gets you your five-coin bet back.

That's 24 winning hands. The other 23 possibilities are losers.

There also are 24 winning possibilities if you start with A-K-Q-J. There's a little less value since there's no chance at a straight flush and there are fewer straights, but there's one more high-card to pair up for a minimal return. On the other four-card royals that include both a 10 and an ace, there are only 21 possible winners and 26 losers.

More four-card royal possibilities include both an ace and a 10 than not, so on balance you'll wind up with a losing hand slightly more than half the time when you start with four cards to a royal flush.

So even with one of the best starting hands in the game, you'll lose more often than you'll win. But that 1 in 47 chance at the 4,000-coin bonanza makes it all worthwhile, and makes the hand a big winner in the long run.

In my e-mail response to Jim, I asked if he'd drawn many royal flushes, and if many had come on one-card draws.

"Oh sure," he replied. "I play for a couple of hours or more, once a week, and I've had my royals. I had three of them last year, and two were on the same day. One royal I drew one card to four parts, just like we're writing about here. One was a two-card draw, and one was a total shock. I had one of those hands with nothing but garbage, no high cards, no flush or straight draws. I threw away all five cards, and got back a royal flush.

"That was really something."