Holding two pair in Jacks or Better: what the draw is worth

You are dealt J-J-5-5-K. The question in video poker is rarely whether to hold the pairs, because the answer is yes. It is how much that hold is worth, and what it is worth compared with the alternatives that look tempting: keep the jacks alone, or keep the jacks and the king. This post puts exact numbers on each option under the 9/6 Jacks or Better paytable, then looks at where the answer moves.

The paytable assumed is 9/6 Jacks or Better, per coin bet: full house 9, flush 6, straight 4, three of a kind 3, two pair 2, jacks or better 1. Expected value (EV) means coins returned per coin bet. You have seen five cards, so 47 are unseen.

The draw: one card, four outs

Hold both pairs and discard the king. Of the 47 unseen cards:

• 2 jacks and 2 fives, 4 cards in all, make a full house, paying 9
• the other 43 leave you with two pair, paying 2

Total: 4 × 9 + 43 × 2 = 36 + 86 = 122 coins across 47 draws. EV = 122 / 47 = 2.60. The full house arrives 4 times in 47, or 8.5%, and the other 91.5% of draws still pay 2.

That second line matters. Nothing you can draw makes the hold worse than what you started with. The floor is 2 coins, every time, and the draw is a free chance at 9. For the bare 4-in-47 odds on their own, see the two pair draw post; this one is about what those odds are worth on a paytable.

Why you never break it for the high pair

The tempting alternative is to keep J-J and draw three cards, since a pair of jacks pays on its own. Counting every one of the 16,215 three-card draws from 47 cards gives a total of 24,980 coins, so EV = 24,980 / 16,215 = 1.54.

That is 1.06 below the 2.60 of the two-pair hold, about 41% less. The two-pair hold is also the safer one. The jacks hold finishes as a bare pair paying 1 on most draws and only improves when it finds trips, two pair or better. Two pair already has the 2 in the bank.

Holding the low pair alone is worse still. Keep the 5s, discard the rest, and the three-card draw is worth about 0.83, because a pair of fives only pays if it improves. The order, best to worst: both pairs (2.60), jacks alone (1.54), fives alone (0.83).

Against three of a kind

Three of a kind and two pair cannot be dealt in the same hand, but the comparison shows where two pair sits on the list. Hold three of a kind with two unmatched cards and draw two from 47. That is worth 4,651 / 1,081 = 4.30 (1,081 two-card draws). It beats two pair because it pays 3 before the draw and two cards to improve it, to a full house or four of a kind.

A pat full house (9) stays untouched, three of a kind (4.30) keeps three cards and two pair (2.60) keeps four. For where these holds sit among all the others, see the Jacks or Better strategy post.

When the answer moves: other variants

The 2.60 belongs to 9/6 Jacks or Better. Other tables pay for the same five cards differently. Video Poker: Jacks or Better offers Jacks or Better, Deuces Wild, Bonus Poker and Double Bonus, and puts a full paytable for each variant on screen, so the first thing to check is the two pair and full house rows.

• Double Bonus (10/7 table) pays 1 for two pair and 10 for a full house. The same draw is worth (4 × 10 + 43 × 1) / 47 = 83 / 47 = 1.77. The hold is worth about two thirds as much as on Jacks or Better, which changes how it compares with other holds on that table. How Bonus Poker and Double Bonus are priced covers the rest.
• Deuces Wild has no two pair row at all: the lowest paying hand is three of a kind, so this hold does not exist there. See the Deuces Wild paytable and its strategy.

A rule to carry away

With two pair, hold four cards and draw one. The draw has 4 winning cards in 47, a floor of 2 and an EV of 2.60 on 9/6 Jacks or Better. No single pair from the same hand gets above 1.54. Check the full house and two pair rows of the paytable before you deal, because those two numbers are what change between variants.