This paper deals with the problem of the optimal choice among building contractor firms tendering for a construction project . The probability and possibility of such firms completing a project in a given period of time is considered. The completion of a job is treated as a fuzzy event The procedure for ascertaining the expected time for the completion of a project at the expected cost of the project is based upon the probability and possibility theory. The selection of a contractor is made upon the minimal expected cost criterion. Following these assumptions a computer program has been created. The example in the paper illustrates the program.