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26<div class="titlepage"><div><div><h5 class="title">
27<a name="math_toolkit.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
28 Sample Sizes for a Binomial Distribution.</a>
29</h5></div></div></div>
30<p>
31 Imagine you have a critical component that you know will fail in 1 in
32 N "uses" (for some suitable definition of "use").
33 You may want to schedule routine replacement of the component so that
34 its chance of failure between routine replacements is less than P%. If
35 the failures follow a binomial distribution (each time the component
36 is "used" it either fails or does not) then the static member
37 function <code class="computeroutput"><span class="identifier">binomial_distibution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_maximum_number_of_trials</span></code>
38 can be used to estimate the maximum number of "uses" of that
39 component for some acceptable risk level <span class="emphasis"><em>alpha</em></span>.
40 </p>
41<p>
42 The example program <a href="../../../../../../example/binomial_sample_sizes.cpp" target="_top">binomial_sample_sizes.cpp</a>
43 demonstrates its usage. It centres on a routine that prints out a table
44 of maximum sample sizes for various probability thresholds:
45 </p>
46<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">find_max_sample_size</span><span class="special">(</span>
47 <span class="keyword">double</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success ratio.</span>
48 <span class="keyword">unsigned</span> <span class="identifier">successes</span><span class="special">)</span> <span class="comment">// Total number of observed successes permitted.</span>
49<span class="special">{</span>
50</pre>
51<p>
52 The routine then declares a table of probability thresholds: these are
53 the maximum acceptable probability that <span class="emphasis"><em>successes</em></span>
54 or fewer events will be observed. In our example, <span class="emphasis"><em>successes</em></span>
55 will be always zero, since we want no component failures, but in other
56 situations non-zero values may well make sense.
57 </p>
58<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>
59</pre>
60<p>
61 Much of the rest of the program is pretty-printing, the important part
62 is in the calculation of maximum number of permitted trials for each
63 value of alpha:
64 </p>
65<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">&lt;</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>
66<span class="special">{</span>
67 <span class="comment">// Confidence value:</span>
68 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</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>
69 <span class="comment">// calculate trials:</span>
70 <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>
71 <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>
72 <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>
73 <span class="comment">// Print Trials:</span>
74 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
75<span class="special">}</span>
76</pre>
77<p>
78 Note that since we're calculating the maximum number of trials permitted,
79 we'll err on the safe side and take the floor of the result. Had we been
80 calculating the <span class="emphasis"><em>minimum</em></span> number of trials required
81 to observe a certain number of <span class="emphasis"><em>successes</em></span> using
82 <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code>
83 we would have taken the ceiling instead.
84 </p>
85<p>
86 We'll finish off by looking at some sample output, firstly for a 1 in
87 1000 chance of component failure with each use:
88 </p>
89<pre class="programlisting">________________________
90Maximum Number of Trials
91________________________
92
93Success ratio = 0.001
94Maximum Number of "successes" permitted = 0
95
96
97____________________________
98Confidence Max Number
99 Value (%) Of Trials
100____________________________
101 50.000 692
102 75.000 287
103 90.000 105
104 95.000 51
105 99.000 10
106 99.900 0
107 99.990 0
108 99.999 0
109</pre>
110<p>
111 So 51 "uses" of the component would yield a 95% chance that
112 no component failures would be observed.
113 </p>
114<p>
115 Compare that with a 1 in 1 million chance of component failure:
116 </p>
117<pre class="programlisting">________________________
118Maximum Number of Trials
119________________________
120
121Success ratio = 0.0000010
122Maximum Number of "successes" permitted = 0
123
124
125____________________________
126Confidence Max Number
127 Value (%) Of Trials
128____________________________
129 50.000 693146
130 75.000 287681
131 90.000 105360
132 95.000 51293
133 99.000 10050
134 99.900 1000
135 99.990 100
136 99.999 10
137</pre>
138<p>
139 In this case, even 1000 uses of the component would still yield a less
140 than 1 in 1000 chance of observing a component failure (i.e. a 99.9%
141 chance of no failure).
142 </p>
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