# The Silence of the Lambs

by Matthew Kidd
February 3, 2015

Bridge text books give the probability of six cards in the opponents’ hands splitting 3-3 as 35.53%, a bit more than one third. This number is calculated as (6,3) × (20,10) / (26,13) where (n,k) denotes the number of different ways to choose k objects (cards) from n objects where the order of the objects is immaterial. Recall that:

(n,k) = n! / ( k! × (n-k)! )     where n! = n × (n-1) × … × 3 × 2 × 1

Here 6 is the number of cards the defenders hold in the suit that splits 3-3 and 20 is the number of cards they hold in all other suits. Each defender needs 3 and 10 cards respectively from these two sets of cards. The plot below shows all the suit split probabilities for six cards.

However, strictly speaking this calculation only applies if you know nothing, i.e. you examine randomly dealt hands. Actual bridge hands include information from the auction. Even opponents who pass throughout the auction are conveying information, the inability to compete. Since it is easier to compete with good shape, opponents who do not compete will on average have flatter hands. How much difference does this make in practice? Below I present simulation results for defender’s distribution in suits where they have six combined cards for various common non-competitive auctions.

## The auction 1♥–2♥–4♥

Let’s begin by considering the auction 1–2–4. I generated 10,000 deals for this auction with the free Wbridge5 program. I used version 4.02 because the latest version 5.1 does not work on my Windows 7 box (crashes at startup). Using Wbridge5 to generate the deals is simpler than using Thomas Andrews’ Deal program as for example was used for to study balancing after one of a suit, because no programming is required. But the convenience comes at the price of not being able to alter the bidding aggressiveness of the opponents as well as slower deal generation. This auction and all that follow are simulated at NV/NV, so the defenders have the best chance of entering the auction. Vulnerable opponents would alter the suit split distributions less.

Here are the results:

# Deals 0-61-52-43-34-25-16-0
2329 0.1 4.7 25.1 40.7 27.1 2.3 0.0
1995 0.4 3.2 23.8 39.9 25.1 7.0 0.7
1851 0.3 4.6 20.3 41.0 27.7 5.7 0.4
# Deals 0-61-52-43-34-25-16-0
2329 -0.6 -2.6 0.9 5.1 2.9 -5.0 -0.8
1995 -0.3 -4.1 -0.5 4.4 0.8 -0.3 -0.0
1851 -0.4 -2.7 -3.9 5.5 3.4 -1.5 -0.4

The defenders only have six card in a suit between them for some of the 10,000 deals. The number of such deals for each suit is shown in the column # Deals. The remaining columns in the first table give the percentage of time that the defender’s cards split each of the seven ways. The 3-3 split probability is emphasized in blue. The second table shows how much each result deviates from what would be predicted if no information were available from the auction.

Observe that the 3-3 split probability is enhanced about 5% is each side suit. This is significant, enough to make one feel more confident about bidding a close vulnerable game at teams after 1–2 when holding a five card side suit on the hope of setting up a putative 5-2 side suit fit.

Also notice that the 2-4 / 4-2 symmetry is broken. LHO is more likely to hold two spades than four spades because with four spades, LHO might have found a takeout double. RHO has a harder time making a takeout double which means RHO’s failure to bid is less revealing.

## Accuracy of the simulation results

Each row in the table above is based on only about 2000 deals. This limits the accuracy of the results. To get a better handle on this issue, here are the results for 10,000 random deals, unconstrained by an auction (generated by Thomas Andrews’ Deal program):

# Deals 0-61-52-43-34-25-16-0
2374 0.7 7.0 23.5 35.4 25.4 7.5 0.6
2381 1.1 6.9 23.1 36.4 25.2 6.6 0.8
2354 0.9 7.1 25.0 33.9 25.0 7.6 0.5
2431 0.7 6.4 25.7 35.8 23.2 7.4 0.9
# Deals 0-61-52-43-34-25-16-0
2374 -0.1 -0.3 -0.7 -0.1 1.2 0.2 -0.2
2381 0.4 -0.4 -1.1 0.8 0.9 -0.7 0.0
2354 0.2 -0.2 0.8 -1.7 0.8 0.4 -0.2
2431 -0.1 -0.9 1.4 0.3 -1.0 0.1 0.1

Results frequently deviate by up to 1% from the results of the exact calculation. This means the digit after the decimal point is mostly insignificant in the percentages shown.

## The auction 1♠–2♠–4♠

# Deals 0-61-52-43-34-25-16-0
2039 0.2 4.9 23.5 41.4 24.8 4.9 0.3
2054 0.2 6.3 22.4 38.2 26.0 6.6 0.2
1983 0.2 4.7 25.3 37.3 27.0 5.3 0.3
# Deals 0-61-52-43-34-25-16-0
2039 -0.6 -2.4 -0.7 5.9 0.6 -2.4 -0.5
2054 -0.5 -0.9 -1.8 2.7 1.8 -0.7 -0.5
1983 -0.6 -2.5 1.1 1.7 2.8 -2.0 -0.5

The results are similar to the 1–2–4 auction above but the 3-3 split enhancement is reduced for the minors.

## The auction 1♣–1♥–2♥–4♥

# Deals 0-61-52-43-34-25-16-0
2688 0.0 0.9 27.0 45.3 25.1 1.6 0.0
733 0.0 5.7 25.9 41.6 23.2 3.4 0.1
2499 0.4 5.4 20.1 44.1 23.6 6.3 0.2
# Deals 0-61-52-43-34-25-16-0
2688 -0.8 -6.4 2.8 9.8 0.9 -5.7 -0.7
733 -0.8 -1.5 1.7 6.1 -1.0 -3.9 -0.6
2499 -0.4 -1.9 -4.1 8.5 -0.6 -1.0 -0.5

Here we see very large enhancement of the 3-3 split probability, almost up 10% for a 3-3 spade break. This is because it is easy for LHO to enter the auction after 1♣; therefore LHO’s failure to do so is more revealing than in auction that begins with a major suit.

Also notice that the diamond suit split results are derived from fewer than 1000 hands. This is because opener has only 2.33 diamonds on average in this auction. The defenders in turn average 7.92 diamonds; holding only six is rare for them, about a 7% chance.

## The auction 1♦–1♥–2♥–4♥

# Deals 0-61-52-43-34-25-16-0
2277 0.0 2.3 27.5 42.6 24.7 2.6 0.1
2689 0.3 5.9 23.2 37.0 25.4 7.7 0.6
1009 0.0 5.5 26.7 39.0 24.0 4.8 0.1
# Deals 0-61-52-43-34-25-16-0
2277 -0.8 -4.9 3.3 7.1 0.5 -4.6 -0.6
2689 -0.5 -1.4 -1.1 1.5 1.2 0.4 -0.2
1009 -0.8 -1.8 2.4 3.5 -0.2 -2.5 -0.7

The enhancement of the 3-3 split probability is reduced compared to the previous auction. It is still significant in spades, the suit that LHO has the easiest time bidding. Clubs must be introduced at the two level and therefore LHO’s failure to bid them is more often for a lack of values than for lack of shape.

As with the previous auction, the results for one minor are derived from comparatively few hands. In this auction, opener’s average club length is only 2.10. The defenders in turn average 7.77 clubs; holding only six is rare for them, about a 10% chance.

## The auction 1♣–1♠–2♠–4♠

# Deals 0-61-52-43-34-25-16-0
1806 0.1 3.8 24.0 44.7 25.9 1.6 0.0
921 0.0 3.7 20.3 41.3 31.5 3.3 0.0
2445 0.0 4.5 17.5 42.0 25.2 10.3 0.4
# Deals 0-61-52-43-34-25-16-0
1806 -0.7 -3.5 -0.2 9.2 1.7 -5.7 -0.8
921 -0.8 -3.6 -3.9 5.7 7.3 -4.0 -0.8
2445 -0.7 -2.8 -6.7 6.5 1.0 3.1 -0.3

This results for this auction are very similar to the auction 1♣–1–2–4 above. The 3-3 split results are virtually indistinguishable from the heart suit auction up to the statistical limits of the simulation. LHO has an easy time entering the auction at LHO’s failure to do so significantly enhances all 3-3 suit split probabilities.

## The auction 1♦–1♠–2♠–4♠

# Deals 0-61-52-43-34-25-16-0
1916 0.3 4.1 25.6 43.0 25.3 1.8 0.0
2579 0.3 4.8 21.2 38.7 26.7 7.8 0.5
1173 0.1 3.1 20.7 42.3 27.6 6.2 0.0
# Deals 0-61-52-43-34-25-16-0
1916 -0.5 -3.1 1.4 7.5 1.0 -5.5 -0.8
2579 -0.4 -2.4 -3.0 3.1 2.5 0.5 -0.3
1173 -0.7 -4.2 -3.5 6.8 3.4 -1.0 -0.8

This results for this auction are similar to the auction 1–1–2–4 above but the 3-3 split probability show greater enhancement for the minor suits.

## The auction 1♣–1♥–1♠–4♠

# Deals 0-61-52-43-34-25-16-0
4356 0.2 7.4 25.8 41.6 23.0 2.0 0.0
220 0.0 3.6 24.5 35.9 31.4 4.5 0.0
2747 0.1 4.2 24.0 40.6 22.6 8.2 0.2
# Deals 0-61-52-43-34-25-16-0
4356 -0.5 0.1 1.6 6.0 -1.2 -5.3 -0.8
220 -0.8 -3.6 0.3 0.4 7.1 -2.7 -0.8
2747 -0.6 -3.0 -0.2 5.1 -1.6 1.0 -0.5

The 3-3 split enhancement is present here. However, the value for diamonds should not be given much consideration because the statistics are low, only 220 hands. The defenders hold an average of 7.86 diamonds on this auction. Only rarely do they have six.

## The auction 1♣–1♦–1♠–4♠

# Deals 0-61-52-43-34-25-16-0
1582 0.1 2.1 28.4 44.4 21.4 3.7 0.0
4427 0.5 8.1 24.9 39.2 23.4 3.6 0.2
2344 0.3 3.7 22.0 44.1 22.4 7.6 0.0
# Deals 0-61-52-43-34-25-16-0
1582 -0.7 -5.1 4.2 8.8 -2.9 -3.6 -0.8
4427 -0.3 0.8 0.7 3.7 -0.8 -3.7 -0.5
2344 -0.5 -3.6 -2.2 8.6 -1.9 0.3 -0.7

The 3-3 split enhancement is large here for both hearts and clubs. The heart suit enhancement is not surprising because both LHO and RHO have an easy time entering the auction with a heart suit.

## Conclusions

• The 3-3 split probability is significantly enhanced for a wide variety of uncontested auctions. When the opponents are silent lambs, they are statistically a few percent closer to the slaughter.
• The enhancement is most significant for spades and when the auction begins with 1♣.
• The 3-3 split probability does not rise as high as 50%, the probability of a finesse, but can rise as high as 45%. (I suspect it may approach 50% when very aggressive opponents are silent but this has not been demonstrated—doing so requires custom simulation code.)

## Get the data

Download a zip file (3 MB) of all the deals (in PBN format) generated for each auction by Wbridge5. Also included is the Perl program silentlamb.pl which was used to compute the statistics and generate the HTML for the tables.