exercise:F4bf733f21: Difference between revisions

Latest revision as of 22:06, 12 June 2024

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 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?

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-<div class="d-none"><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
-\newcommand{\NA}{{\rm NA}}
+will occur (see [[exercise:9e83c3ee89 |Exercise]]).
-\newcommand{\mat}[1]{{\bf#1}}
-\newcommand{\exref}[1]{\ref{##1}}
+<ul style="list-style-type:lower-alpha"><li> Show that the probability that, in <math>m</math> trials, there are no successes
-\newcommand{\secstoprocess}{\all}
-\newcommand{\NA}{{\rm NA}}
-\newcommand{\mathds}{\mathbb}</math></div>
-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 [[guide:4f3a4e96c3#exer 1.1.5 ||Exercise]].]]).
-<ul><li> Show that the probability that, in <math>m</math> trials, there are no successes
 is <math>(1 - 1/n)^m</math>.
 </li>