Shape versus strength
In bridge, each of four players receives 13 cards. Two things describe a hand: its strength, usually counted in high-card points, and its shape, the way the 13 cards are spread across spades, hearts, diamonds and clubs. The shape is written as four numbers in descending order, so a hand with five spades, four hearts, two diamonds and two clubs is a 5-4-2-2 hand.
Shape matters because the same strength plays very differently depending on where the cards sit. A hand with a long suit can win tricks with small cards in that suit once the opponents have run out, while a hand with no long suit relies on high cards and on the cards partner holds. The probabilities of the shapes are fixed by the deck, and they are worth knowing because the most common patterns are not the ones people tend to expect.
The common shapes
The number of 13-card hands is 635,013,559,600, and each shape accounts for a share of them. Counting the hands that fit each pattern, using the same counting methods that give the total, produces these approximate figures:
| Shape | Approximate share of hands |
|---|---|
| 4-4-3-2 | 21.6% |
| 5-3-3-2 | 15.5% |
| 5-4-3-1 | 12.9% |
| 5-4-2-2 | 10.6% |
| 4-3-3-3 | 10.5% |
| 6-3-2-2 | 5.6% |
| 6-4-2-1 | 4.7% |
| 6-3-3-1 | 3.5% |
| 5-5-2-1 | 3.2% |
| 4-4-4-1 | 3.0% |
| 7-3-2-1 | 1.9% |
The eleven shapes in the table cover about 93% of deals. The remainder is spread over many rarer shapes, including hands with a void (no cards in one suit). Voids in total occur in only around one hand in twenty.
The numbers show something people often find surprising. The “flattest” shape, 4-3-3-3, is only about one hand in ten, and the single most common shape is 4-4-3-2. One reason is that a shape can be assigned to the four suits in different ways. For 4-3-3-3 only the choice of which suit holds the four matters, giving 4 arrangements, while 4-4-3-2 has 12 (which suits hold the fours, which holds the three). More arrangements means more hands.
Balanced shapes (4-3-3-3, 4-4-3-2 and 5-3-3-2, all with no singleton or void) together account for roughly 47.6% of hands, based on the figures above (10.5 + 21.6 + 15.5). Roughly half of hands are therefore balanced, and the other half have a singleton, a void or a long suit.
How opponents’ cards tend to split
Bidding tells you something about your own side’s cards, but play depends heavily on how the missing cards in a suit are divided between the two opponents. These are the figures before any other information is used, and they come from counting how the remaining 26 cards can be shared out:
- Four cards missing: split 2-2 about 40.7% of the time, 3-1 about 49.8%, and 4-0 about 9.6%.
- Five cards missing: split 3-2 about 67.8%, 4-1 about 28.3%, and 5-0 about 3.9%.
- Six cards missing: split 4-2 about 48.4%, 3-3 about 35.5%, 5-1 about 14.5%, and 6-0 about 1.5%.
Each group sums to 100% within rounding. The pattern is the same as for the hands themselves: an even split is the single most likely result with an odd number of missing cards (3-2), but with an even number missing, the even split is not the most likely. With four missing, an uneven 3-1 beats 2-2, and with six missing, 4-2 beats 3-3. A common summary is that an odd number of missing cards tends to divide as evenly as it can, while an even number tends not to, though these are tendencies, not guarantees.
How this guides bidding and play
Players use these percentages as rough guides. In bidding, shape helps decide whether a hand suits a no-trump contract or a trump suit. Balanced hands usually fit no-trump, while hands with a long suit or a short suit often favour a trump contract, because a short suit lets a player trump losing cards. Partners combine their shapes through the auction, and a combined holding of eight or more cards in a suit is commonly treated as a good basis for trumps. This is general convention, and systems differ.
In the play, declarer counts the missing cards and compares the likely splits with what the contract needs. If a suit needs to split 3-2 to produce the extra tricks, the figure above shows the plan works about two times in three before any other clues. The opponents’ bidding, their opening leads and the cards that fall can change that, which is why the figures are called starting estimates. That updating is an example of how new information changes the odds, as in conditional probability.
Limits of the percentages
Bridge is often described as a mix of luck versus skill, and these percentages sit on the luck side of that line: they describe the deal, not the player’s judgment. A 68% chance is a long-run frequency across many deals, not a promise about this deal. Any single hand can split badly, and a plan that is the best percentage play can still fail. Percentages also assume the missing cards are dealt randomly, so knowledge from the auction can make a specific split more or less likely than the table suggests.
Hand-shape arithmetic is a good example of how dealing a fixed deck produces regular, predictable proportions over many deals, without telling you anything certain about the next one.