<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Estimating Sample Sizes for a Binomial Distribution.</title> <link rel="stylesheet" href="../../../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../../../index.html" title="Math Toolkit"> <link rel="up" href="../binom_eg.html" title="Binomial Distribution Examples"> <link rel="prev" href="binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution"> <link rel="next" href="../neg_binom_eg.html" title="Negative Binomial Distribution Examples"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="binom_conf.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../binom_eg.html"><img src="../../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../../index.html"><img src="../../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../neg_binom_eg.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section" lang="en"> <div class="titlepage"><div><div><h6 class="title"> <a name="math_toolkit.dist.stat_tut.weg.binom_eg.binom_size_eg"></a><a class="link" href="binom_size_eg.html" title="Estimating Sample Sizes for a Binomial Distribution."> Estimating Sample Sizes for a Binomial Distribution.</a> </h6></div></div></div> <p> Imagine you have a critical component that you know will fail in 1 in N "uses" (for some suitable definition of "use"). You may want to schedule routine replacement of the component so that its chance of failure between routine replacements is less than P%. If the failures follow a binomial distribution (each time the component is "used" it either fails or does not) then the static member function <code class="computeroutput"><span class="identifier">binomial_distibution</span><span class="special"><>::</span><span class="identifier">find_maximum_number_of_trials</span></code> can be used to estimate the maximum number of "uses" of that component for some acceptable risk level <span class="emphasis"><em>alpha</em></span>. </p> <p> The example program <a href="../../../../../../../../example/binomial_sample_sizes.cpp" target="_top">binomial_sample_sizes.cpp</a> demonstrates its usage. It centres on a routine that prints out a table of maximum sample sizes for various probability thresholds: </p> <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">find_max_sample_size</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success ratio. </span> <span class="keyword">unsigned</span> <span class="identifier">successes</span><span class="special">)</span> <span class="comment">// Total number of observed successes permitted. </span><span class="special">{</span> </pre> <p> The routine then declares a table of probability thresholds: these are the maximum acceptable probability that <span class="emphasis"><em>successes</em></span> or fewer events will be observed. In our example, <span class="emphasis"><em>successes</em></span> will be always zero, since we want no component failures, but in other situations non-zero values may well make sense. </p> <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span> </pre> <p> Much of the rest of the program is pretty-printing, the important part is in the calculation of maximum number of permitted trials for each value of alpha: </p> <pre class="programlisting"><span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span> <span class="special">{</span> <span class="comment">// Confidence value: </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span> <span class="comment">// calculate trials: </span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">binomial</span><span class="special">::</span><span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span> <span class="identifier">successes</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span> <span class="comment">// Print Trials: </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="special">}</span> </pre> <p> Note that since we're calculating the maximum number of trials permitted, we'll err on the safe side and take the floor of the result. Had we been calculating the <span class="emphasis"><em>minimum</em></span> number of trials required to observe a certain number of <span class="emphasis"><em>successes</em></span> using <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code> we would have taken the ceiling instead. </p> <p> We'll finish off by looking at some sample output, firstly for a 1 in 1000 chance of component failure with each use: </p> <pre class="programlisting">________________________ Maximum Number of Trials ________________________ Success ratio = 0.001 Maximum Number of "successes" permitted = 0 ____________________________ Confidence Max Number Value (%) Of Trials ____________________________ 50.000 692 75.000 287 90.000 105 95.000 51 99.000 10 99.900 0 99.990 0 99.999 0 </pre> <p> So 51 "uses" of the component would yield a 95% chance that no component failures would be observed. </p> <p> Compare that with a 1 in 1 million chance of component failure: </p> <pre class="programlisting">________________________ Maximum Number of Trials ________________________ Success ratio = 0.0000010 Maximum Number of "successes" permitted = 0 ____________________________ Confidence Max Number Value (%) Of Trials ____________________________ 50.000 693146 75.000 287681 90.000 105360 95.000 51293 99.000 10050 99.900 1000 99.990 100 99.999 10 </pre> <p> In this case, even 1000 uses of the component would still yield a less than 1 in 1000 chance of observing a component failure (i.e. a 99.9% chance of no failure). </p> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p> Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="binom_conf.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../binom_eg.html"><img src="../../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../../index.html"><img src="../../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../neg_binom_eg.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>