The supply of women into senior management has changed little despite well-intentioned efforts. We argue that the biggest effect is from supply-side factors that inhibit females' decision to enter competitions: Women are under-confident about winning, men are over-confident; women are more risk averse than men in some settings; and, most importantly, women shy away from competition. In order to change the conditions under which this is the case, this paper proposes a radical idea. It is to use a particular form of random selection of candidates to increase the supply of women into management positions. We argue that selective randomness would encourage women to enter tournaments; offer women 'rejection insurance'; ensure equality over time; raise the standard of candidates; reduce homophily to improve diversity of people and ideas; and lessen 'the chosen one' factor. We also demonstrate, using Jensen's inequality from applied mathematics, that random selection can improve organizational efficiency.