ceph/src/boost/libs/math/doc/html/math_toolkit/stat_tut/weg/binom_eg/binomial_coinflip_example.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Binomial Coin-Flipping Example</title>
   5 <link rel="stylesheet" href="../../../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
   7 <link rel="home" href="../../../../index.html" title="Math Toolkit 2.5.1">
   8 <link rel="up" href="../binom_eg.html" title="Binomial Distribution Examples">
   9 <link rel="prev" href="../binom_eg.html" title="Binomial Distribution Examples">
  10 <link rel="next" href="binomial_quiz_example.html" title="Binomial Quiz Example">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="../binom_eg.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="binomial_quiz_example.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.stat_tut.weg.binom_eg.binomial_coinflip_example"></a><a class="link" href="binomial_coinflip_example.html" title="Binomial Coin-Flipping Example">Binomial
  28           Coin-Flipping Example</a>
  29 </h5></div></div></div>
  30 <p>
  31             An example of a <a href="http://en.wikipedia.org/wiki/Bernoulli_process" target="_top">Bernoulli
  32             process</a> is coin flipping. A variable in such a sequence may be
  33             called a Bernoulli variable.
  34           </p>
  35 <p>
  36             This example shows using the Binomial distribution to predict the probability
  37             of heads and tails when throwing a coin.
  38           </p>
  39 <p>
  40             The number of correct answers (say heads), X, is distributed as a binomial
  41             random variable with binomial distribution parameters number of trials
  42             (flips) n = 10 and probability (success_fraction) of getting a head p
  43             = 0.5 (a 'fair' coin).
  44           </p>
  45 <p>
  46             (Our coin is assumed fair, but we could easily change the success_fraction
  47             parameter p from 0.5 to some other value to simulate an unfair coin,
  48             say 0.6 for one with chewing gum on the tail, so it is more likely to
  49             fall tails down and heads up).
  50           </p>
  51 <p>
  52             First we need some includes and using statements to be able to use the
  53             binomial distribution, some std input and output, and get started:
  54           </p>
  55 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  56   <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">binomial</span><span class="special">;</span>
  57
  58 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
  59   <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span>  <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>  <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span>
  60 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
  61   <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span>
  62
  63 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
  64 <span class="special">{</span>
  65   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Using Binomial distribution to predict how many heads and tails."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  66   <span class="keyword">try</span>
  67   <span class="special">{</span>
  68 </pre>
  69 <p>
  70             See note <a class="link" href="binomial_coinflip_example.html#coinflip_eg_catch">with the catch block</a>
  71             about why a try and catch block is always a good idea.
  72           </p>
  73 <p>
  74             First, construct a binomial distribution with parameters success_fraction
  75             1/2, and how many flips.
  76           </p>
  77 <pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// = 50% = 1/2 for a 'fair' coin.</span>
  78 <span class="keyword">int</span> <span class="identifier">flips</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span>
  79 <span class="identifier">binomial</span> <span class="identifier">flip</span><span class="special">(</span><span class="identifier">flips</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span>
  80
  81 <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">4</span><span class="special">);</span>
  82 </pre>
  83 <p>
  84             Then some examples of using Binomial moments (and echoing the parameters).
  85           </p>
  86 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"From "</span> <span class="special">&lt;&lt;</span> <span class="identifier">flips</span> <span class="special">&lt;&lt;</span> <span class="string">" one can expect to get on average "</span>
  87   <span class="special">&lt;&lt;</span> <span class="identifier">mean</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" heads (or tails)."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  88 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Mode is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mode</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  89 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviation is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  90 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"So about 2/3 will lie within 1 standard deviation and get between "</span>
  91   <span class="special">&lt;&lt;</span>  <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">flip</span><span class="special">))</span>  <span class="special">&lt;&lt;</span> <span class="string">" and "</span>
  92   <span class="special">&lt;&lt;</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">flip</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="string">" correct."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  93 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Skewness is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  94 <span class="comment">// Skewness of binomial distributions is only zero (symmetrical)</span>
  95 <span class="comment">// if success_fraction is exactly one half,</span>
  96 <span class="comment">// for example, when flipping 'fair' coins.</span>
  97 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Skewness if success_fraction is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">flip</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span>
  98   <span class="special">&lt;&lt;</span> <span class="string">" is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Expect zero for a 'fair' coin.</span>
  99 </pre>
 100 <p>
 101             Now we show a variety of predictions on the probability of heads:
 102           </p>
 103 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"For "</span> <span class="special">&lt;&lt;</span> <span class="identifier">flip</span><span class="special">.</span><span class="identifier">trials</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">" coin flips: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 104 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting no heads is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 105 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting at least one head is "</span> <span class="special">&lt;&lt;</span> <span class="number">1.</span> <span class="special">-</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 106 </pre>
 107 <p>
 108             When we want to calculate the probability for a range or values we can
 109             sum the PDF's:
 110           </p>
 111 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 0 or 1 heads is "</span>
 112   <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// sum of exactly == probabilities</span>
 113 </pre>
 114 <p>
 115             Or we can use the cdf.
 116           </p>
 117 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 0 or 1 (&lt;= 1) heads is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 118 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 9 or 10 heads is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">9</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 119 </pre>
 120 <p>
 121             Note that using
 122           </p>
 123 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 9 or 10 heads is "</span> <span class="special">&lt;&lt;</span> <span class="number">1.</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">8</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 124 </pre>
 125 <p>
 126             is less accurate than using the complement
 127           </p>
 128 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting 9 or 10 heads is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">8</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 129 </pre>
 130 <p>
 131             Since the subtraction may involve <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">cancellation
 132             error</a>, where as <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">8</span><span class="special">))</span></code>
 133             does not use such a subtraction internally, and so does not exhibit the
 134             problem.
 135           </p>
 136 <p>
 137             To get the probability for a range of heads, we can either add the pdfs
 138             for each number of heads
 139           </p>
 140 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of between 4 and 6 heads (4 or 5 or 6) is "</span>
 141   <span class="comment">//  P(X == 4) + P(X == 5) + P(X == 6)</span>
 142   <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">5</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">6</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 143 </pre>
 144 <p>
 145             But this is probably less efficient than using the cdf
 146           </p>
 147 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of between 4 and 6 heads (4 or 5 or 6) is "</span>
 148   <span class="comment">// P(X &lt;= 6) - P(X &lt;= 3) == P(X &lt; 4)</span>
 149   <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">6</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 150 </pre>
 151 <p>
 152             Certainly for a bigger range like, 3 to 7
 153           </p>
 154 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is "</span>
 155   <span class="comment">// P(X &lt;= 7) - P(X &lt;= 2) == P(X &lt; 3)</span>
 156   <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">7</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 157 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 158 </pre>
 159 <p>
 160             Finally, print two tables of probability for the <span class="emphasis"><em>exactly</em></span>
 161             and <span class="emphasis"><em>at least</em></span> a number of heads.
 162           </p>
 163 <pre class="programlisting"><span class="comment">// Print a table of probability for the exactly a number of heads.</span>
 164 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting exactly (==) heads"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 165 <span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">successes</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">successes</span> <span class="special">&lt;=</span> <span class="identifier">flips</span><span class="special">;</span> <span class="identifier">successes</span><span class="special">++)</span>
 166 <span class="special">{</span> <span class="comment">// Say success means getting a head (or equally success means getting a tail).</span>
 167   <span class="keyword">double</span> <span class="identifier">probability</span> <span class="special">=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">);</span>
 168   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">successes</span> <span class="special">&lt;&lt;</span> <span class="string">"     "</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>
 169     <span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">&lt;&lt;</span> <span class="string">" or 1 in "</span> <span class="special">&lt;&lt;</span> <span class="number">1.</span> <span class="special">/</span> <span class="identifier">probability</span>
 170     <span class="special">&lt;&lt;</span> <span class="string">", or "</span> <span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">*</span> <span class="number">100.</span> <span class="special">&lt;&lt;</span> <span class="string">"%"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 171 <span class="special">}</span> <span class="comment">// for i</span>
 172 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 173
 174 <span class="comment">// Tabulate the probability of getting between zero heads and 0 upto 10 heads.</span>
 175 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting upto (&lt;=) heads"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 176 <span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">successes</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">successes</span> <span class="special">&lt;=</span> <span class="identifier">flips</span><span class="special">;</span> <span class="identifier">successes</span><span class="special">++)</span>
 177 <span class="special">{</span> <span class="comment">// Say success means getting a head</span>
 178   <span class="comment">// (equally success could mean getting a tail).</span>
 179   <span class="keyword">double</span> <span class="identifier">probability</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">flip</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">);</span> <span class="comment">// P(X &lt;= heads)</span>
 180   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">successes</span> <span class="special">&lt;&lt;</span> <span class="string">"        "</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">left</span>
 181     <span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">&lt;&lt;</span> <span class="string">" or 1 in "</span> <span class="special">&lt;&lt;</span> <span class="number">1.</span> <span class="special">/</span> <span class="identifier">probability</span> <span class="special">&lt;&lt;</span> <span class="string">", or "</span>
 182     <span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">*</span> <span class="number">100.</span> <span class="special">&lt;&lt;</span> <span class="string">"%"</span><span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 183 <span class="special">}</span> <span class="comment">// for i</span>
 184 </pre>
 185 <p>
 186             The last (0 to 10 heads) must, of course, be 100% probability.
 187           </p>
 188 <pre class="programlisting"><span class="special">}</span>
 189 <span class="keyword">catch</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&amp;</span> <span class="identifier">e</span><span class="special">)</span>
 190 <span class="special">{</span>
 191   <span class="comment">//</span>
 192 </pre>
 193 <p>
 194             <a name="coinflip_eg_catch"></a>It is always essential to include try
 195             &amp; catch blocks because default policies are to throw exceptions on
 196             arguments that are out of domain or cause errors like numeric-overflow.
 197           </p>
 198 <p>
 199             Lacking try &amp; catch blocks, the program will abort, whereas the message
 200             below from the thrown exception will give some helpful clues as to the
 201             cause of the problem.
 202           </p>
 203 <pre class="programlisting">  <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
 204     <span class="string">"\n"</span><span class="string">"Message from thrown exception was:\n   "</span> <span class="special">&lt;&lt;</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 205 <span class="special">}</span>
 206 </pre>
 207 <p>
 208             See <a href="../../../../../../example/binomial_coinflip_example.cpp" target="_top">binomial_coinflip_example.cpp</a>
 209             for full source code, the program output looks like this:
 210           </p>
 211 <pre class="programlisting">Using Binomial distribution to predict how many heads and tails.
 212 From 10 one can expect to get on average 5 heads (or tails).
 213 Mode is 5
 214 Standard deviation is 1.581
 215 So about 2/3 will lie within 1 standard deviation and get between 4 and 6 correct.
 216 Skewness is 0
 217 Skewness if success_fraction is 0.5 is 0
 218
 219 For 10 coin flips:
 220 Probability of getting no heads is 0.0009766
 221 Probability of getting at least one head is 0.999
 222 Probability of getting 0 or 1 heads is 0.01074
 223 Probability of getting 0 or 1 (&lt;= 1) heads is 0.01074
 224 Probability of getting 9 or 10 heads is 0.01074
 225 Probability of getting 9 or 10 heads is 0.01074
 226 Probability of getting 9 or 10 heads is 0.01074
 227 Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6562
 228 Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6563
 229 Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is 0.8906
 230
 231 Probability of getting exactly (==) heads
 232 0      0.0009766  or 1 in 1024, or 0.09766%
 233 1      0.009766   or 1 in 102.4, or 0.9766%
 234 2      0.04395    or 1 in 22.76, or 4.395%
 235 3      0.1172     or 1 in 8.533, or 11.72%
 236 4      0.2051     or 1 in 4.876, or 20.51%
 237 5      0.2461     or 1 in 4.063, or 24.61%
 238 6      0.2051     or 1 in 4.876, or 20.51%
 239 7      0.1172     or 1 in 8.533, or 11.72%
 240 8      0.04395    or 1 in 22.76, or 4.395%
 241 9      0.009766   or 1 in 102.4, or 0.9766%
 242 10     0.0009766  or 1 in 1024, or 0.09766%
 243
 244 Probability of getting upto (&lt;=) heads
 245 0         0.0009766  or 1 in 1024, or 0.09766%
 246 1         0.01074    or 1 in 93.09, or 1.074%
 247 2         0.05469    or 1 in 18.29, or 5.469%
 248 3         0.1719     or 1 in 5.818, or 17.19%
 249 4         0.377      or 1 in 2.653, or 37.7%
 250 5         0.623      or 1 in 1.605, or 62.3%
 251 6         0.8281     or 1 in 1.208, or 82.81%
 252 7         0.9453     or 1 in 1.058, or 94.53%
 253 8         0.9893     or 1 in 1.011, or 98.93%
 254 9         0.999      or 1 in 1.001, or 99.9%
 255 10        1          or 1 in 1, or 100%
 256 </pre>
 257 </div>
 258 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 259 <td align="left"></td>
 260 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 261       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 262       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 263       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 264         Distributed under the Boost Software License, Version 1.0. (See accompanying
 265         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>)
 266       </p>
 267 </div></td>
 268 </tr></table>
 269 <hr>
 270 <div class="spirit-nav">
 271 <a accesskey="p" href="../binom_eg.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="binomial_quiz_example.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
 272 </div>
 273 </body>
 274 </html>