When it comes to running a lottery and keeping it honest in appearance as well as fact, most governments leave nothing to chance except the winners. That is why, as we noted in this space two months ago, most states run their games directly instead of farming them out to private operators; they also have tight rules barring even anyone with friends who have ties to illegal gambling or the underworld from working for the official operations. But in the D.C. lottery, where the standards are not nearly so stringent, the relationships of the operation's key players is curious, and the coincidences so sheer that they border on transparent.

So far nothing illegal has been discovered. Still, from the outset there have been questions from all corners--Mayor Barry himself, the D.C. Council and firms competing for lottery business--about contract bidding and awarding, and about the people managing the games, including one who was convicted 11 years ago of operating an illegal lottery, maintaining a gambling establishment and possessing illegal numbers slips. Now there is yet another intriguing account of relationships involving other key players:

FBI agents investigating the lottery operations reportedly are trying to determine if there is a connection between lottery board member Lillian C. Wiggins' cash purchase of a $4,000 fur coat and a $5,000 check cashed the same day by the principal owner of the company that does the advertising for the city's lottery games.

Teresa Suter, the majority stockholder in the advertising firm--LaMancha, Inc.--met Mrs. Wiggins at a furrier's on the same day that Mrs. Wiggins bought the coat and helped her pick it out, Mrs. Wiggins has told a reporter. She said they and another LaMancha official were to have lunch that day, which they did. She also says she paid for the coat with her own money. An attorney for LaMancha says Teresa Suter did not give any of the $5,000 to Mrs. Wiggins.

Coincidence? Until or unless proven otherwise, yes. The same goes for a report that the FBI is examining an allegation that another board member, Jerry Cooper, was given a Cadillac by an official of a company that received a lottery board contract. Mr. Cooper says he told the agents that last winter he drove a blue Cadillac owned by his sister, but that he owns and normally drives a 1976 Dodge.

It may be coincidental, too, that Mayor Barry has been less than pleased with the independence that Mrs. Wiggins and Mr. Cooper have shown as his appointees to the board. The mayor has nominated replacements for them, in fact.

We continue to believe that the findings of all the investigators are critical, not just to the future of legalized gambling in this city but also to the political franchise and financial integrity of the city itself. In the meantime, every party involved in the lottery operations should go to great lengths to avoid even the appearance of improper relationships that can destroy public trust.