A: Yes they are truly random. This is a common misperception since many players are often used to relatively "flat" hands produced in typical shuffle and deal games. We use the industry standard utility Bridge Composer which does not provide any options for attenuating or manipulating hand distributions. In addition, we are using the open source utlity "Big Deal" (an option inside Bridge Composer) to generate the actual hands. These hands are added to our library and NEVER touched thereafter.
All tests fell within the statistical expectations of their outcomes, in my judgment.
For the testing, I used data from July 23, 2013 through August 20, 2014. This equates to 394 games, 14,184 deals, and 56,736 separate hands.
The tests were designed to explore the various theories I've heard about how the hands might be skewed.
For an example of testing the actual frequencies in the hand record data against the theoretical frequencies, let's look at the number of hands with 7+ cards in any suit.
Such a test might be a proxy for the hands seeming to have wild distribution or too many long suits. The theoretical percent probability that a hand will have a suit with 7+ cards is 4.03%.
Further, the statistical formulas that account for the variability across samples tell us that 90% of the samples of 14,184 random bridge hands will have somewhere
between 3.76% and 4.30% hands with 7+ cards in a suit. Looking at the North hands in the data, we find there are 589 hands with a 7+ suit of the 14,184 total hands or 4.15% -
well within the 90% boundaries calculated above. For East, South, and West, the percentage of hand with a 7+ suit are 3.98%, 3.97%, and 4.25% respectively.
At the sample sizes we have, the statistical tests are extremely sensitive. Let's suppose AS AN EXAMPLE ONLY, that the deal generator was incorrectly putting out a hands with
7+ cards in a suit 4.8% of the time. On average, that's about one extra 7+ hand every 130 boards or about one every four to five games.
In spite of the small increase in frequency of 7+ hands, the test above would detect the increase about 99.8% of the time.
If the increase in frequency were as large as 1-2 hands per round - the range where we are likely to notice it - this test could not possibly have missed it.
This is the rationale for using large samples. Even very small deviations from expected frequencies can be reliably detected.
So, when none are found, one could infer that meaningful deviations are not present.
Here are the types of tests done and the rationale for each.
1. Firstly, I looked at the distribution of the shapes of the hands, e.g., 4333, 4432, 4441, 5332, etc.
No statistically significant differences were found relative to the theoretical distribution for bridge hands.
This test is a proxy for weird distributions
2. Looked at the distribution of voids and singletons over the groups of 36 boards.
There were no deviations from statistical expectations for these tests. These are proxies for weird distribution. The specific tests were:
a. Number of voids per 36 board session
b. Number of singletons per 36 board session
3. For singletons, looked at the frequency in each of the four directions, for each suit and for each denomination.
The frequency of singletons should be statistically equal in each of the four directions, four suits, and 13 card values.
That is what the tests showed. These tests are for claims like, “Singleton aces are too common” or “I always get the stiff kings.”
4. Looked at the position of the honors to see if there was any deviation from the expectation that (absent other information) the missing honor is equally likely to be on the left or right.
The test found no statistical difference from the expected distribution for the positions of adjacent honors.
This test is a proxy for “all of the finesses are lose or are fixed.”
5. Looked at the suit breaks for a partnership, especially for shorter suits. This investigates whether the opponents' trump are split as expected by the probability distribution.
They do. This is a proxy test for “I always get terrible trump splits.”
6. Looked at the presence of each card in the hands. That is, is each hand equally likely to hold the ace of spades or whatever.
No deviations were found overall, within the 36 board sets, or in the pattern for the presence/absence of cards.
This is a proxy test for “I seem to get {a given card} a lot.”
7. Looked at the distribution of HCP overall and within the 36 board sets. No deviations found.
8. Looked at the average HCP and the variation in the HCP's by hand in the 36 board sets. Everything fit with statistical expectations.
9. Did the same as #8 with the combined HCP for the partnership. Everything fit with statistical expectations.
This is the proxy for the claim, “they always get enough points to go to game contracts, we get part scores.”
10. Looked for any cycling, patterns or correlations in the HCP within a set of 36 boards. None found.
To summarize, the set of tests I ran found no violations of the theoretical frequencies and expectations for randomly dealt bridge hands.
The TCG - The Common Game Hand Distribution from: 04/01/2013 - 04/30/2013:
First 24 Boards Hand Pattern Occurrence Probability
Pattern *ACBL % TCG Diff
4-4-3-2 21.55% 21.11 0.44
5-3-3-2 15.52 16.49 -0.97
5-4-3-1 12.93 12.85 0.08
5-4-2-2 10.58 9.90 0.68
4-3-3-3 10.53 10.24 0.29
6-3-2-2 5.64 5.45 0.19
6-4-2-1 4.70 5.14 -0.44
6-3-3-1 3.45 4.06 -0.61
5-5-2-1 3.17 3.75 -0.58
4-4-4-1 2.99 2.85 0.14
7-3-2-1 1.88 1.63 0.25
6-4-3-0 1.33 1.46 -0.13
5-4-4-0 1.24 1.11 0.13
5-5-3-0 0.90 0.73 0.17
Other 6 Card 1.43 1.39 0.04
Other 7 Card 1.65 1.35 0.30
8-10 card suits 0.6 0.49 0.11
First 24 Boards Hand Pattern Occurrence Probability from: 01/01/2014 - 01/31/2014
Pattern ACBL % TCG Diff
4-4-3-2 21.55% 20.23 1.32
5-3-3-2 15.52 15.63 -0.11
5-4-3-1 12.93 13.51 -0.58
5-4-2-2 10.58 10.55 0.03
4-3-3-3 10.53 10.82 -0.29
6-3-2-2 5.64 6.18 -0.54
6-4-2-1 4.70 4.40 0.30
6-3-3-1 3.45 3.46 -0.01
5-5-2-1 3.17 3.29 -0.12
4-4-4-1 2.99 3.19 -0.20
7-3-2-1 1.88 1.68 0.20
6-4-3-0 1.33 1.44 -0.11
5-4-4-0 1.24 1.28 -0.04
5-5-3-0 0.90 1.01 -0.11
Other 6 Card 1.43 1.11 0.32
Other 7 Card 1.65 1.51 0.14
8-10 card suits 0.6 0.71 -0.11
*ACBL Distribution Source