Holding low pairs in Double Double Bonus

On a flight to Las Vegas, I was whiling away the time practicing 9/6 Double Double Bonus Poker on my laptop. That's not one of my favorite Vegas games, because with the 9/6 pay table, Double Double Bonus returns a mere 99.0% with expert play. You can do better than that in Las Vegas, with games including full-pay Deuces Wild (100.8) and 10/7/5 Double Bonus (100.2) returning more than 100% to experts.

In the Midwest, though, 9/6 Double Double Bonus is often one of the best games in the house, so it's in the mix of games I practice regularly.

As I practiced, the woman seated next to me watched a little and chatted. Time after time, she'd see me hold a low pair instead of a single high card, and she wanted to know if there was a method to my madness.

She introduced herself as Janet, and said video poker was her favorite game. "Are you sure that's the right play?" she asked.

The software I was using, Bob Dancer presents WinPoker, would warn me of any incorrect plays, I explained. Holding the low pair is mathematically the better play.

"I always thought holding the low pair was better in Jacks or Better, but that in Double Double Bonus, where you only get your money back on two pair, you were better off to hold the high card."

I knew at that point that she'd played a fair amount of video poker. Changing strategy with different games and pay tables is the kind of thing video poker players fret about.

I explained to her that you win more HANDS if you hold just the high card, but you win more MONEY if you hold the low pair.

"More money is good," she said, laughing. "But why do you win more money? Aren't most of your winners going to be two-pair hands? In Double Double Bonus, those hands pay the same amount as when you pair up a high card. It's not like Jacks or Better, where you get 2-for-1 on two pair."

It is true that two-pair hands are the most frequent winners when you start with a low pair. But you also get three-of-a-kind and four-of-a-kind A LOT more often starting with a pair than a single high card. And in Double Double Bonus, those quads are huge.

I asked if she wanted to see the full breakdown. She did, so I clicked on "analyze hand" and defined the hand as a queen of hearts, 7s of spades and diamonds, a 10 of clubs and a 5 of spades.

If you hold just the queen, there are 178,365 possible four-card draws. The majority, 111,413 of them, bring no payoff. The next biggest share, 45,324, get us our money back — five coins on a five-coin maximum bet — with a pair of jacks or better. (The biggest share of those are queens, of course, but we will also sometimes find pairs of jacks, kings and aces in our draws.) We'll also get five-coin returns on 9,033 possible two-pair hands, 15-coin returns on 4,177 three of a kinds, 20 coins on 573 straights, 25 coins on 492 flushes, 45 on 297 full houses, 250 on 49 four of a kinds consisting of 5s through kings, 400 on each of three four of a kinds consisting of 2s, 3s or 4s, and 800 on the one four-ace hand.

Average all that out, and you'll find that you've had winners on 37.5% of hands and averaged a 2.18-coin return for every five coins wagered.

What if you hold the pair of 7s instead? That narrows the possibilities. There are only 16,125 possible three-card draws. Again, the majority, 11,559 hands, bring no return. There are no flush or straight possibilities, so our paying hands will include 5 coins on each of 2,592 two-pair hands, 15 on each of 1,854 three of a kinds, 45 on each of 165 full houses and 250 on each of 45 four-seven hands.

We have winners on only 28.3% of all possibilities, but our average return leaps to 3.62 coins per five coins wagered.

Why? Because a much higher proportion of our winners get us more than our money back. Hold just the Queen, and 54,257 of 66,952 possible winners, or 81%, are either high pairs or two pairs that just get our money back. Hold the 7s, and the 2,592 two-pairs in our 4,566 winners mean we get five-coin returns on only 56.8% of our winners.

We win less often when we hold the pair, but we also win more money.

I asked Janet if she was now sufficiently bored. She probably didn't expect a lecture on numbers when she booked the flight.

She laughed. "Sometimes small questions get big answers," she said. "But I learned something. I'll be holding those pairs."