Seventy Percent Sessions

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by Matthew Kidd

Though I’ve come close, I don’t think I’ve ever had a 70+% session. By the time I was good enough to beat up on weaker players that badly, I had moved on to better games where it seems much harder to have a 70+% sessions even for the strongest pairs. Not surprisingly, most of the 70+% sessions I read about involve pairs I don’t know playing in podunk clubs or pairs I do know who are either clearly sandbagging or just playing in a weak field to prevent a half table. I’ve never felt that a 70+% session was comparable to winning an open pairs event of 20 or so tables at a sectional, let alone winning one at a regional.

I was resolved to ignore the whole 70+% game hoopla until I read Annetta Patrick’s Santa Barbara unit update in the May 2014 issue of the Contract Bridge Forum (page 15) where she wrote:

You may recall that I once likened a 70% game to getting struck by lightning, or shooting a Hole-In-One. Well, I’ve been researching the probabilities of this phenomenon, but have come up empty. Would you “Math Minds” out there lend me a hand? And even more interesting, what is the probability of having four 70% games within a five day time frame? A bottle of wine to anyone who can find some authoritative proof on this matter.

I wasn’t gunning for the bottle of wine, but in the back of my mind I could hear Annetta thinking in her best Carl Sagan voice that the odds must be, “trillions and trillions to one.” Off-hand I figured it was a lot more likely and that I had better put out an estimate before the week of 70% sessions got established as a modern miracle, ranked up there with faces of the Virgin Mary in tree sap and on cheese sandwiches.

To make any progress answering Annetta’s question it is necessary to know the probability of any pair achieving a 70+% session in any single session. This is likely to depend on (1) the field strength; (2) the number of tables; (3) the diversity of skill in the field; and (4) the variability of the pairs in the field, particularly for the better pairs. It seems that it would be harder to have a big session in a strong field because there are too many opponents doing sensible things and empirical observation backs this up. The number of tables matters because, more pairs means more pairs that might exceed 70% even if each pair has only a small chance to do so. Conversely, a very small session of only three tables might help because even a tie for top is 1.5 out of 2 matchpoints, a 75% board; good play and a some luck could result in a big session. Regarding diversity, a weak field with one or two strong pairs, for example when a director and a regular partner play in a weak, but still technically open game, should offer greater chances for a 70+% session. Finally, the variability of the pairs matters somewhat because for example a pair that averages 55% ± 8% is considerably more likely than a pair that averages 55% ± 5% to have an upward swing carrying them over 70%.

Still it is best to have actual numbers. To determine the probability of a 70+% game in fields of varying strength, I modified my Payoff Matrix software to calculate the field strength and also to tally the number of pairs achieving a 70% session and the fraction of sessions in which at least one pair had achieved a 70+% result. These features are now included in the 1.0.8 release of the software. Here are the results for several sets of sessions, each containing at least a year of data.

Field strength for each session
Set of Sessions Mean
Pair ≥ 70%
%Prob Any
Pair ≥ 70%
Esplanade Club (all 2013 limited) 86 42 1.45 17.86
Charlotte Bridge Studio (≤ 750 MP UL, 2013) 120 72 0.51 7.97
Eastlake Club (Wed Morning open, 2013) 400 150 0.33 6.25
Esplanade Club (all 2013 open) 1160 530 0.39 6.91
Charlotte Bridge Studio (all 2013 open) 1600 800 0.23 5.07
Soledad Club (Thu Eve open, 2008–2012) 2000 1240 0.13 2.15
San Diego Unit Games (2012–2013 open) 2800 1750 0.14 4.55
La Jolla Unit Games (2007–2014 open) 2700 1600 0.12 4.52

For each set of sessions, both the mean and geometric mean of the field strength are shown. I have argued previously that the geometric mean is a better indicator of field strength than the mean. In a geometric mean, the mean is taken in a logarithmic manner. For example, the geometric mean of two players with 10 and 1000 MP respectively, is 100 MP not 505 MP. In a weak field with one or only a few strong pairs, the geometric mean is distorted far less by these pairs than the arithmetic mean. This is readily apparent in the Eastlake Wednesday Morning game where the geometric mean is much less than the arithmetic mean.

The second to last column gives the probability that a randomly chosen pair will achieve a 70+% result in any session. Observe that this percentage falls dramatically as the field strength increases. It is roughly ten times more likely that a random pair will have a 70+% game in the weak Esplanade limited games than in the either of the unit games or the Soledad Thursday evening game. Having a 70+% session in any game is far more common than being struck by lightening. If lightening strikes were nearly so common, membership would take a big blow.

The last column gives that probability that at least one pair will achieve a 70+% result in any session. This also falls off with field strength, but no nearly as dramatically because it is also a function of the average number of tables. Open game are typically larger than limited games and unit games are bigger yet than typical open games. Here is a plot of the data above.

Plot of field strength versus probability of a randomly chosen pair having a 70+% session (left) and any pair having a 70+% session (right)

To assign an accurate probability to the run of 70+% games in the Santa Barbara we need data from each game to determine the probability that any pair will achieve a 70+% session in that game. Annetta has not sent me any ACBLscore game files yet. Nonetheless the full text from the Contract Bridge Forum indicates at the first 70+% result was in a 299er game and it is reasonable to assume the next two were also because no mention is made of open pairs until the final 70+% game on March 31. Based on the Charlotte Bridge Studio limited game set in the table above, it is reasonable to use 8% as an estimate that any pair will achieve an 70+% session for the games under consideration.

The probability of four straight 70+% sessions, not necessarily by the same pair, is therefore roughly 8%4 = (0.08)4 ≈ 4 ×10-5 or about 24,000:1. But were the 70+% sessions consecutive? Annetta’s text doesn’t say but the answer affects the probability calculation considerably. To see why, consider a coin toss results observed as heads five times. If this happened in five tosses, you might be mildly surprised. If it happened in ten tosses you would hardly think twice about it.

More generally, we need to use the binomial distribution:

Pr(k) = (n,k) pk (1-p)n-k

where (n,k) = n! / (k! × (n-k)!) = number of ways to choose k object from n objects.

Here p is the probability of something happening once and Pr(k) is the probability of it happening k times given n opportunities to happen. For coin tossing, p is ½. For these 70+% sessions p ≈ 0.08.

If the club has two sessions per day, there are 10 total sessions over the five day period over which the four 70+% games occurred. So k=4 and n=10 and we have:

Pr(4) = (10,4) (0.08)4 (0.92)6 = 210 × (0.08)4 (0.92)6 = 0.0052 = 0.52%

Now the probability has risen to half a percent or 200:1 against. This isn’t rare at all.

On the other hand, if there is only one session per day, n=5 and the odds are about 5000:1.

Probability that a specific partnership has a 70+% session

Individual partnerships differ in both their strength and variability where the latter attribute was examined in the article Partnership Variability. To answer Annetta’s question where we don’t care which partnership has a 70+% session, we relied on aggregate results from games of different strength. But as an aside, let’s consider the probability that a specific partnership has a 70+% session.

First consider an average partnership with the typical 6% variability, i.e. a partnership with a 50% ± 6% average. Results for partnerships are roughly Gaussian (bell-curve shaped). For this partnership, a 70% game is (70 − 50) / 6 = 3.3 standard deviations from their average result. Using the so-called error function, we can calculate the probability of a 70+% game for this partnership as erfc(3.33/sqrt(2)) / 2 = 0.04% or roughly once every 50 years if they play together once a week and never improve. Next consider a strong partnership with the typical 6% standard deviation, i.e. a partnership with a 55% ± 6% average. A similar calculation yields 0.6%, meaning 70+% games will be more than ten times as frequent. Finally, consider a strong partnership with a high 8% standard deviation. This time the calculation yields 3%, five times higher yet than the equally strong partnership with less variability. This partnership should have a 70% game once or twice a year if they play once per week.

I caution that the assumption of a Gaussian distribution may not be valid all the way out to results as high as 70%. So while in principle, the probability that at least one pair achieves a 70+% game might be calculated strictly from a knowledge of the means and standard deviations of each partnership in the event as determine over say 50 or 100 session of the same game, it is safer to simply tally up 70+% results and use that as the input to calculations as I have done above for Annetta’s question.