Financial General Bankshares Inc., Washington's third-largest banking company, posted a 22 percent increase in profits for 1981, despite an 11 percent drop in fourth-quarter earnings.

The holding company for the First American Banks and others earned just over $20 million ($2.75 a share) for the year, after deducting securities losses of $47,000, compared with a net of $16.4 million ($2.24) after securities transaction losses of $145,000 in 1980.

For the fourth quarter, however, Financial General's earnings slipped to $4.1 million (57 cents) from $4.7 million (63 cents). Chairman B. Francis Saul II attributed the rough fourth quarter to a decline in short-term interest rates that reduced interest income by less than 3 percent. For the year, net interest income was up 14 percent and non-interest income gained 26 percent.

Deposits rose 7 percent to $2.2 billion and assets were up 8 percent to $2.6 billion at year's end.

A. H. Robins Co., the big Richmond-based drug maker, posted a 74 percent increase in profits last year with earnings of $44.2 million ($1.77 a share) from sales of $450 million, compared with $25.5 million ($1.01) on sales of $432.3 million the year before.

Sales of Robins' present operations increased by 14 percent, but the company sold off two unprofitable subsidiaries during the year.