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If you have an infinitesimal chance of succeeding at something, but you do it an infinite amount of times, what is the probability that you succeed at least once?

Jacob Regan
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    that's a question of limits of functions, for example if the chance of the thing not happening is $(1 - \frac{1}{n})^n$ as n heads off to infinity we get $\frac{1}{e}$ so $1 - \frac{1}{e}$ could be an answer – Cato Apr 19 '17 at 09:50
  • In classical probability theory, a probability is a real number between $0$ and $1$, $0$ and $1$ included. Hence there is no such thing as an infinitesimally small (but non-zero) chance. What precisely do you mean? – Bib-lost Apr 19 '17 at 09:54
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    @Bib-lost It was pretty much just a something I wondered after watching a YouTube video about infinitesimals, I guess if there is no such thing as an infinitesimally small chance then that is the answer – Jacob Regan Apr 19 '17 at 09:58
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    It is presumably possible to do probability with non-classical real numbers, but I'm not sure it would help much in practice. – Thomas Andrews Apr 19 '17 at 10:01
  • @ThomasAndrews, your comment reveals a vast ignorance of the subject. Abraham Robinson's approach relies on classical logic, contrary to what your comment suggests. – Mikhail Katz Apr 19 '17 at 10:23

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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.

Mikhail Katz
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