LARO77 Three Pictures: How the Hands Score and How Rare the Top Hand Is

Three Pictures is a three-card comparing game, and it belongs with the live casino tables this site covers, in the family of card rules you can work out on paper. Each player gets three cards, picture cards count zero, and a hand of three picture cards tops the ranking. This page gives you the card values, the commonly described ranking and the arithmetic for how rare each kind of hand is. Open the game’s own rules screen before you play, because tables differ in the details.

How Three Pictures works

The game is a quick comparison of two things, points and the top hand. The steps below are the commonly described round, so treat them as the frame and check the table’s own screen for its version.

1. Every player and the dealer receive three cards.
2. Each hand is added up using the card points in the next table.
3. The points total is the last digit of the sum, so 14 counts as 4 and 20 counts as 0.
4. A hand of three picture cards ranks above every points hand.
5. The hands are compared and the higher one wins.

Card points: what each card is worth

Picture cards are the Jack, Queen and King. They are worth nothing toward the total, which is why the game gets its name.

A worked example: King, 7 and 8 add to 15, so the hand scores 5. Queen, Jack and 9 scores 9, because the two picture cards add nothing. Ace, 9 and 10 adds to 20 and scores 0.

The commonly described ranking

The ranking is short, which keeps the game fast. Read it from the top down.

1. Three picture cards: the top hand, above all points hands.
2. 9 points.
3. 8 points, then 7, and so on down to 1 point.
4. 0 points, the lowest hand.

How a table settles two equal hands, and how it treats a hand with two picture cards, differs between rule sets. The game’s own rules screen is the place to read those two details.

How often three picture cards arrive

The deck holds 12 picture cards in 52. For three picture cards in a row, multiply the chance at each card, because each card taken leaves one fewer picture card and one fewer card overall.

(12/52) x (11/51) x (10/50) = 1,320 / 132,600 = 0.00995

That is 0.995%, about 1 hand in 100.45. The same answer comes from counting hands: 220 three-card groups of pictures divided by the 22,100 possible three-card groups in the deck. The order in which cards arrive does not change it, so every seat has the same chance.

How many picture cards a hand holds

Counting the 22,100 possible three-card groups from one deck by how many picture cards they hold gives this split.

The working for the middle rows: one picture card is 12 picture cards times the 780 pairs of the 40 other cards, which is 9,360. Two picture cards is 66 pairs of pictures times 40 other cards, which is 2,640. Counts like these are why a hand with at least one picture card shows up in more than half of all deals.

Points for the other hands

Take the 21,880 groups that are not three pictures, count the Ace as 1, and score each by the last digit. The counts split almost evenly across the ten point values.

Each point value is close to one tenth of the non-picture hands. A hand scoring 9 is therefore common, and the top hand is the only rare one. Read how a soft hand counts the ace for a different counting rule on the same deck, and how the small road is drawn for a scoreboard that records results without predicting them.

What the odds do and do not tell you

Rarity tells you how seldom a hand arrives. It does not tell you what a bet on that hand costs, because that depends on the payout the table applies. Open the table’s own paytable and rules screen, then use what house edge meaning covers at a table to turn a payout and a probability into an expected cost.

1. Find the payout for the bet on the paytable.
2. Multiply the chance of winning by the return per unit staked.
3. Subtract the result from 1 to get the cost of each unit staked.

Every figure on this page is an example of arithmetic from a 52-card deck. A multi-deck shoe or a different rule set changes the counts, so the game’s own rules screen has the final word.