[awm]

Fairly frequently people make statements about what "works" and what "doesn't." Often these are based on some vague generalities and there don't seem to be a whole lot of facts other than a few made up example hands or remembered experiences to back them up (or to refute them). My interest in stuff like double dummy solvers is to try to get at least some basis in fact for these things.

One statement I've seen a lot of people make is that, when you hold a 4441 hand, you are "very likely to find a fit." This is used as justification for all sorts of conventions and bids, with mini-roman 2♦ being perhaps the most atrocious of them. I've seen a lot of people say this on these forums. Fortunately, this one can be resolved without a double dummy solver. Here's what I found out:

In general, with no assumptions about distribution, your side has:

7-card fit only: 15.66%
8-card fit: 45.83%
9-card fit: 28.07%
More: 10.44%

Obviously the odds change with more information. If you are 4441, then:

7-card fit only: 16.60%
8-card fit: 49.56%
9-card fit: 26.81%
More: 7.03%

Holding 4441 makes you less likely to have an 8+ card fit than you would be on a random hand!

Of course, this is biased somewhat because hands with 7-card suits (for example) are extremely likely to produce an 8-card fit. So let's compare to some other frequent hand patterns:

If you are 5422, then:

7-card fit only: 15.38%
8-card fit: 47.05%
9-card fit: 28.30%
More: 9.27%

If you are 5332, then:

7-card fit only: 17.41%
8-card fit: 48.19%
9-card fit: 26.15%
More: 8.25%

From this, one can conclude that while the chances of a fit when you are 4441 are somewhat better than they would be if you have a balanced hand, the odds are not particularly good relative to mildly unbalanced hands with five-card suits. In addition, we can observe that the chance for a "super-fit" when you are 4441 is not particularly good.

In addition, combined with some of Tysen's earlier data, we can conclude that 4441 hands actually have good playing strength and that the apparent strength is not some sort of illusion generated by the high probability of having a good fit.

All of this data is based upon generating a million random hands, so the percentages will not be exactly correct, although the error should be quite small. I'll leave it to the statisticians to generate standard deviations and error rates if they care.

[000002]

GOOD

[Gerben42]

With my Precision partner I play 2♦ as 3-suited with short ♦. We don't play it because the bid is so great, because we always find a fit, yada yada yada. Instead the main reason is to fix a hole in the system.

So this bid works in the sense that it makes 1♦ real.

[Chamaco]

Endless discussion...
Some players (many in Italy) bid 4441 hands treating them as balanced....

Then another bunch of players immediately object "what if pard xfers in your singleton playing you for a doubleton?"...