[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Assume that the probability of a “success” on a single experiment with [math]n[/math] outcomes is [math]1/n[/math]. Let [math]m[/math] be the number of experiments necessary to make it a favorable bet that at least one success will occur (see [[guide:4f3a4e96c3#sec 1.1 |Exercise.]]).

• Show that the probability that, in [math]m[/math] trials, there are no successes is [math](1 - 1/n)^m[/math].
• (de Moivre) Show that if [math]m = n \log 2[/math] then
  [[math]] \lim_{n \to \infty} \left(1 - \frac1n \right)^m = \frac12\ . [[/math]]
  Hint:
  [[math]] \lim_{n \to \infty} \left(1 - \frac1n \right)^n = e^{-1}\ . [[/math]]
  Hence for large [math]n[/math] we should choose [math]m[/math] to be about [math]n \log 2[/math].
• Would DeMoivre have been led to the correct answer for de Méré's two bets if he had used his approximation?