Choosing a hold in video poker: the highest average wins.

Thirty-two ways to play five cards.

Each card is either held or thrown away, so five cards can be played 2 × 2 × 2 × 2 × 2 = 32 ways. The fewer you keep, the more draws there are to average over.

Two numbers in that table are worth a second look. The draws across all 32 holds add up to 2,598,960, which is every five-card hand in the deck exactly once. And on 9/6 Jacks or Better the best play keeps two cards on 63.54% of deals, almost always a pair or two high cards. Throwing all five away is right on only 3.25%.

“Best play on” counts all 2,598,960 deals of 9/6 Jacks or Better, each held the best of its 32 ways.

How one hold is priced.

Take A♠ K♠ Q♠ J♠ 3♣ and hold the four spades. One card comes from the 47 you have not seen, so there are exactly 47 possible draws. Sort them by the hand they make, add up what they pay, divide by 47.

Price your own hold.

Pick a hand or deal one, then tap the cards you would keep. Every one of the 32 holds is priced in your browser over every possible draw, and yours is set against the best.

Returns are for every 1 bet at five coins a hand, over every possible draw from the 47 cards left. The percentages are how often each hold finishes on each hand.

Most hands are easy. Some are not.

For every one of the 2,598,960 deals we measured how far the best hold is ahead of the next best. That lead is the least a wrong choice can cost on that hand, for every 1 you bet.

9/6 Jacks or Better. The lead is the best hold’s return minus the next-best hold’s, for every 1 bet; holds that are exactly equal count as one.

Close calls are common: on 27.41% of deals the next-best hold is within 0.05 of a bet. Each one costs little, but they add up, and they are the hands that feel cannot settle. The averages are less forgiving: always taking the next-best hold would cost an extra 0.1947 for every 1 you bet, or 19.47% of everything, against a house edge of 0.46%.

“Close calls”: the next-best hold within 0.05 of a bet. “Draw five” and “keep two”: how often the best play throws away every card, or keeps exactly two. Deuces Wild throws everything away far more often because high cards are worth nothing there: with no deuce, no pair and no draw, there is nothing worth keeping.

Why every strategy chart is a ranked list.

Nobody prices 32 holds between hands. A strategy chart does it once, in advance: it sorts the kinds of holding you can have (four to a royal, a high pair, four to a flush and so on) by what they return, and you hold the highest one in your hand.

That is all a chart is. It comes from exactly the sums above, run over every deal, and then compressed into a list short enough to learn. Each game needs its own list, because the pay table moves the sums. Here is one hand in three games:

In Jacks or Better two pair pays 2 for 1, which makes a high pair valuable, so the kings win. Double Double Bonus pays two pair only 1 for 1, and the royal draw edges ahead. In Deuces Wild a pair of kings pays nothing at all unless it improves, so the royal draw wins easily. Same cards, two different right answers.

The exact cards matter as well as the game. In Double Double Bonus most three-card royals still lose to a high pair; king-queen-jack of one suit, as here, is the only one that usually beats it. A chart has to list exceptions like that one as well as the main order.

Every figure is what the hold returns for every 1 bet, over every possible draw.

What the sums keep saying.

Run the sums on enough hands and the same patterns turn up. We checked each of these five on every hand of its kind in 9/6 Jacks or Better; only one has an exception, and it is noted. Each is shown on one hand, with both choices priced.

Questions about holding.