exercise:5caece39e2: Difference between revisions
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(Created page with "<div class="d-none"><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></div> A coin is tossed until the first time a head turns up. If this occurs on the <math>n</math>th toss and <math>n</math> is odd you win <math>2^n/n</math>, but if <math>n</math> is even then you lose <math>2^n/n</math>. Then if your expected winnin...") |
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A coin is tossed until the first time a head turns up. If this occurs on the <math>n</math>th toss and <math>n</math> is odd you win <math>2^n/n</math>, but if <math>n</math> is even | |||
this occurs on the <math>n</math>th toss and <math>n</math> is odd you win <math>2^n/n</math>, but if <math>n</math> is even | |||
then you lose | then you lose | ||
<math>2^n/n</math>. Then if your expected winnings exist they are given by the convergent series | <math>2^n/n</math>. Then if your expected winnings exist they are given by the convergent series | ||
Latest revision as of 17:19, 14 June 2024
A coin is tossed until the first time a head turns up. If this occurs on the [math]n[/math]th toss and [math]n[/math] is odd you win [math]2^n/n[/math], but if [math]n[/math] is even then you lose [math]2^n/n[/math]. Then if your expected winnings exist they are given by the convergent series
[[math]]
1 - \frac 12 + \frac 13 - \frac 14 +\cdots
[[/math]]
called the alternating harmonic series. It is tempting to say that this should be the expected value of the experiment. Show that if we were to do this, the expected value of an experiment would depend upon the order in which the outcomes are listed.