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The Question: How much do distribution points matter when going for a 4S contract?
The Short Answer: They matter, but to incorporate distribution effectively you have to always count them or never count them.
Context: In this analysis distribution points are counted as 1 point for every non-spade suit one holds that has two cards, 2 for a 1 card holding, 3 for a void.
25 HCP 4 Spade Game
25 HCP 4 Spade Game
The table to the left shows the number of games & odds of making that game by the combined number of spades held & combined distribution points for all 25 HCP hands. Interestingly, only 1 hand won out of 17 in the 8 card fit with no distribution. Also useful is the 7 card fit is at nearly 50% once there are 4 distribution points or more.
Below are the tables for 22 through 28 High Card Points
Comments of the Tables Above
I started with 25 High Card Points because when not factoring in distribution points 25 HCP with an 8 card major fit is the cutoff for a more likely than not game. More on that here. Normally I show the number of data points to validate the percentages. Obviously one doesn't have a 100% chance of making game with a 27 HCP point holding, 11 card spade fit, and 3 distribution points. Out of 400,000 hands there were only 3 holding like that. But the reason I mention this is that the number of HCP & spade fit effect distribution points. Here is the point directly, from this data, if one has 25 HCP, and 8 spades, you only have a 5.1% chance of having fewer than 2 distribution points. One is most likely to have 3 distribution points.
For those who like stretching their holding to make game (useful in team games), it's interesting to see that 8 card fits with 4 distribution points & at least 22 HCP get there more often than not.
Distribution of Distribution
Distribution of Distribution
The header may read weird by I thought it was too funny not to use. The table to the right is a histogram of the combined distribution points for 399,991 hands. The bottom of the graph is edited from what it shows in the analysis because of the way Excel deals with bin width. The most common holding is 2 distribution points. When considering both 2 & 3 disruption point hands, one has a 54.12% chance of falling in that range according to this data set.
Table of Chances for Total Points
Table of Chances for Total Points
This table combines HCP with Distribution points & checks chances of making game against the spade fit. In this case a normal 8 card fit requires 27 total points to have above a 50% chance of making game. This fits well with previously reported data. If one needs 25 HCP & 8 spades to get the same result AND we know the average partnership holding has 2 distribution points it stands to reason this table shows you need 27 total points.
A final note: in regular play it's tough to sort out what's what unless you & partner have a solid agreement. A bidding sequence may go 1S > Pass > you? and you're looking at a 3-3-3-4 holding and 12 HCP. You think you have enough for game per the tables above but what if partner opened and 11 count with a singleton in clubs? As of now I don't know the best way to sort this.
Comments & Methods
I think for most folks the above is what they care about, the results. The raw data used in the analysis was generated using a Python program that generated 200,000 hands. From each of the hands the Double Dummy analysis was taken using the DDStable Python library. Lastly, the hand shape, suits, and high card points were extracted from each hand.
There's a human factor of Bridge not accounted for here. A wise Bridge player once said, "Defense is hard." Often times I've heard from players much better than myself that the Double Dummy analysis of a hand favors the defending side. It assumes the defense finds the perfect lead. While that factor will certainly change ones real life percentages, I hold it doesn't materially affect these results but is worth acknowledging.
From the 200,000 hands, 400,000 actual attacks were analyzed. I took every combination regardless of shape & HCP and pulled the resulting makeable No Trump contracts & Major suited contracts possible according to Double Dummy. Even though a contract can be played from either side, the lowest value was used to determine what contract was makeable. For example, if North could make 3NT if they were declarer but South could only make 2NT as declarer the whole hand was deemed to make only 2NT.
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