You are dealt the ace, king, queen and jack of hearts and the seven of hearts. That is a flush, paying 6 for each coin bet with no draw needed. In 9/6 Jacks or Better the correct play is to discard the seven and keep four cards. This post counts the draw exactly, then checks a made straight, a high pair, a straight-flush draw and a pat straight flush.
The paytable assumed is 9/6 Jacks or Better at five coins, per coin bet: royal flush 800, straight flush 50, four of a kind 25, 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 and the draw takes one of them.
Made flush against four to a royal
Hold A-K-Q-J of hearts. The 47 unseen cards divide like this:
- 1 card (the ten of hearts) makes a royal flush, paying 800
- 7 other hearts make a flush, paying 6 each
- 3 off-suit tens make a straight, paying 4 each
- 12 aces, kings, queens and jacks make jacks or better, paying 1 each
- the other 24 pay nothing
Total: 800 + 7 × 6 + 3 × 4 + 12 × 1 = 866 coins across 47 draws. The EV is 866 / 47 = 18.43, against a certain 6. Breaking the flush is worth 18.43 / 6 = 3.07 times as much.
Where the value comes from
The royal is 1 draw in 47, or 2.1%, yet it supplies 800 of the 866 coins, 92.4% of the EV. Everything else adds up to 66 / 47 = 1.40, and 24 of the 47 draws (51.1%) pay nothing. You give up a sure 6 for a draw that fails just over half the time.
The royal prize only has to be big enough to carry that. Without it the hold needs a royal worth x where (66 + x) / 47 = 6, so x = 282 − 66 = 216. The paytable pays 800, which is why the call is not close. See the royal flush share of the return for the same row seen from the other side.
Made straight and high pair
A straight pays only 4. Take A-K-Q-J of hearts with the ten of spades. Keep the four hearts. Now 2 tens can make a straight, since the ten of spades is gone, and 8 other hearts make a flush: 800 + 8 × 6 + 2 × 4 + 12 × 1 = 868 coins, or 868 / 47 = 18.47 against 4.
If the fifth card pairs one of the four, as with the jack of spades, the pair is worth 24,915 / 16,215 = 1.54 (jacks, three cards drawn from 47). The four to a royal is worth 871 / 47 = 18.53, with 11 jacks-or-better outs instead of 12. Discard the pair.
The straight-flush draw case
Hold 10-J-Q-K of hearts and two cards help: the ace of hearts for a royal, and the nine of hearts for a straight flush paying 50. With an off-suit nine as the fifth card (a made straight), the draw is:
- 1 royal flush: 800
- 1 straight flush: 50
- 7 other hearts make a flush: 42
- 5 off-suit aces and nines make a straight: 20 (one nine is the card you threw away)
- 9 jacks, queens and kings: 9
That is 921 coins, so EV = 921 / 47 = 19.60. With a low heart as the fifth card instead (a made flush) the sum is 800 + 50 + 36 + 24 + 9 = 919, an EV of 19.55. The flush version is 53 coins above A-K-Q-J with a low heart (919 − 866), or 1.13 per coin bet. The extra card is worth a little, not a lot, because 10-J-Q-K has fewer jacks-or-better outs (9 against 12).
The exception: keep a pat straight flush
Deal 9-10-J-Q-K of hearts and the hand is already a straight flush paying 50. Discard the nine to chase the royal and the draw is 800 + 7 × 6 + 6 × 4 + 9 × 1 = 875 coins, an EV of 875 / 47 = 18.62. Keep the 50. A pat royal flush and four of a kind (25) also stay. Both sit above the draw in the list from the Jacks or Better strategy post.
Three to a royal is different
J-Q-K of hearts with two low hearts is a made flush, and the three-card draw is worth 1,555 / 1,081 = 1.44 against a pat 6. Keep the flush. The royal needs four cards held before it outweighs a made hand. For the smaller holds, see low pair or four to a flush.
These numbers belong to 9/6 Jacks or Better at five coins. At one to four coins the royal pays 250 per coin bet. The break is still correct, as (66 + 250) / 47 = 6.72 is above 6, but the margin is thinner.
A rule to carry away
Four to a royal beats a made flush, a made straight and a high pair. Only a pat straight flush, four of a kind or royal flush is worth more. Three to a royal does not. Count to four, then break the hand.
Video Poker: Jacks or Better plays Jacks or Better offline, shows a full paytable on screen for each variant, and has strategy hints you can turn on to see which cards to hold while you learn. Chips are free and have no cash value.