Gender differences in choosing to enter competitions are an important cause of the leaky pipeline for women in leadership roles and represent a considerable waste of human resources. We used an incentivized laboratory ex-periment to evaluate whether the introduction of random elements alters the gender gap in competitiveness. We found that focal random selection from a preselected pool removes the difference in competitiveness between men and women and does not dilute the qualifications of the entrants. The percentage of women who took part in competitions was nearly triple, and that of high-ability women double, with focal random selection compared to selection in pure performance competitions. In contrast, the behavior of men remained largely unchanged. Focal random selection closes the gender gap in competitiveness and can substantially enlarge the pool of high-performing women who apply for top jobs.