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<title>Estimating Sample Sizes for a Binomial Distribution.</title>
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<div class="section math_toolkit_dist_stat_tut_weg_binom_eg_binom_size_eg">
<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>
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Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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