Probability based on infinitesimals is a well-established subject with a vast publication list. Indeed it is possible to make sense of statements like "infinitely many trials, each with infinitesimal chance of winning, can have appreciable overall probability". This approach was pioneered by de Mises and formalized by Edward Nelson. The approach is "radically elementary" in the sense that it does not require complex developments in measure theory as other approaches to probability on continuous spaces do. Just to mention a recent article, you could consult this.