It is debated whether the pervasive intergenic transcription from eukaryotic genomes has functional significance or simply reflects the promiscuity of RNA polymerases. We approach this question by comparing chance promoter activities with the expression levels of intergenic regions in the model eukaryote Saccharomyces cerevisiae. We build a library of over 10 strains, each carrying a 120-nucleotide, chromosomally integrated, completely random sequence driving the potential transcription of a barcode. Quantifying the RNA concentration of each barcode in two environments reveals that 41-63% of random sequences have significant, albeit usually low, promoter activities. Therefore, even in eukaryotes, where the presence of chromatin is thought to repress transcription, chance transcription is prevalent. We find that only 1-5% of yeast intergenic transcriptions are unattributable to chance promoter activities or neighboring gene expressions, and these transcriptions exhibit higher-than-expected environment-specificity. These findings suggest that only a minute fraction of intergenic transcription is functional in yeast.