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| <div class="titlepage"><div><div><h6 class="title"> |
| <a name="math_toolkit.dist.stat_tut.weg.binom_eg.binomial_coinflip_example"></a><a class="link" href="binomial_coinflip_example.html" title="Binomial Coin-Flipping Example"> |
| Binomial Coin-Flipping Example</a> |
| </h6></div></div></div> |
| <p> |
| </p> |
| <p> |
| An example of a <a href="http://en.wikipedia.org/wiki/Bernoulli_process" target="_top">Bernoulli |
| process</a> is coin flipping. A variable in such a sequence may |
| be called a Bernoulli variable. |
| </p> |
| <p> |
| </p> |
| <p> |
| This example shows using the Binomial distribution to predict the |
| probability of heads and tails when throwing a coin. |
| </p> |
| <p> |
| </p> |
| <p> |
| The number of correct answers (say heads), X, is distributed as a |
| binomial random variable with binomial distribution parameters number |
| of trials (flips) n = 10 and probability (success_fraction) of getting |
| a head p = 0.5 (a 'fair' coin). |
| </p> |
| <p> |
| </p> |
| <p> |
| (Our coin is assumed fair, but we could easily change the success_fraction |
| parameter p from 0.5 to some other value to simulate an unfair coin, |
| say 0.6 for one with chewing gum on the tail, so it is more likely |
| to fall tails down and heads up). |
| </p> |
| <p> |
| </p> |
| <p> |
| First we need some includes and using statements to be able to use |
| the binomial distribution, some std input and output, and get started: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span> |
| <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> |
| |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <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> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> |
| |
| <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> |
| <span class="special">{</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Using Binomial distribution to predict how many heads and tails."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="keyword">try</span> |
| <span class="special">{</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| See note <a class="link" href="binomial_coinflip_example.html#coinflip_eg_catch">with the catch block</a> |
| about why a try and catch block is always a good idea. |
| </p> |
| <p> |
| </p> |
| <p> |
| First, construct a binomial distribution with parameters success_fraction |
| 1/2, and how many flips. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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><span class="keyword">int</span> <span class="identifier">flips</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span> |
| <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> |
| |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Then some examples of using Binomial moments (and echoing the parameters). |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"From "</span> <span class="special"><<</span> <span class="identifier">flips</span> <span class="special"><<</span> <span class="string">" one can expect to get on average "</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="string">" heads (or tails)."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Mode is "</span> <span class="special"><<</span> <span class="identifier">mode</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard deviation is "</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"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"So about 2/3 will lie within 1 standard deviation and get between "</span> |
| <span class="special"><<</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"><<</span> <span class="string">" and "</span> |
| <span class="special"><<</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"><<</span> <span class="string">" correct."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Skewness is "</span> <span class="special"><<</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="comment">// Skewness of binomial distributions is only zero (symmetrical) |
| </span><span class="comment">// if success_fraction is exactly one half, |
| </span><span class="comment">// for example, when flipping 'fair' coins. |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Skewness if success_fraction is "</span> <span class="special"><<</span> <span class="identifier">flip</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> |
| <span class="special"><<</span> <span class="string">" is "</span> <span class="special"><<</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">flip</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Expect zero for a 'fair' coin.</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Now we show a variety of predictions on the probability of heads: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"For "</span> <span class="special"><<</span> <span class="identifier">flip</span><span class="special">.</span><span class="identifier">trials</span><span class="special">()</span> <span class="special"><<</span> <span class="string">" coin flips: "</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting no heads is "</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"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting at least one head is "</span> <span class="special"><<</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"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| When we want to calculate the probability for a range or values we |
| can sum the PDF's: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting 0 or 1 heads is "</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">+</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"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// sum of exactly == probabilities</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Or we can use the cdf. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting 0 or 1 (<= 1) heads is "</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">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting 9 or 10 heads is "</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">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"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Note that using |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting 9 or 10 heads is "</span> <span class="special"><<</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"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| is less accurate than using the complement |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting 9 or 10 heads is "</span> <span class="special"><<</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"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Since the subtraction may involve <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">cancellation |
| 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> does not use such a subtraction |
| internally, and so does not exhibit the problem. |
| </p> |
| <p> |
| </p> |
| <p> |
| To get the probability for a range of heads, we can either add the |
| pdfs for each number of heads |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of between 4 and 6 heads (4 or 5 or 6) is "</span> |
| <span class="comment">// P(X == 4) + P(X == 5) + P(X == 6) |
| </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">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"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| But this is probably less efficient than using the cdf |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of between 4 and 6 heads (4 or 5 or 6) is "</span> |
| <span class="comment">// P(X <= 6) - P(X <= 3) == P(X < 4) |
| </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">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"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Certainly for a bigger range like, 3 to 7 |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is "</span> |
| <span class="comment">// P(X <= 7) - P(X <= 2) == P(X < 3) |
| </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">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"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Finally, print two tables of probability for the <span class="emphasis"><em>exactly</em></span> |
| and <span class="emphasis"><em>at least</em></span> a number of heads. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="comment">// Print a table of probability for the exactly a number of heads. |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting exactly (==) heads"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <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"><=</span> <span class="identifier">flips</span><span class="special">;</span> <span class="identifier">successes</span><span class="special">++)</span> |
| <span class="special">{</span> <span class="comment">// Say success means getting a head (or equally success means getting a tail). |
| </span> <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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">successes</span> <span class="special"><<</span> <span class="string">" "</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">probability</span> <span class="special"><<</span> <span class="string">" or 1 in "</span> <span class="special"><<</span> <span class="number">1.</span> <span class="special">/</span> <span class="identifier">probability</span> |
| <span class="special"><<</span> <span class="string">", or "</span> <span class="special"><<</span> <span class="identifier">probability</span> <span class="special">*</span> <span class="number">100.</span> <span class="special"><<</span> <span class="string">"%"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> <span class="comment">// for i |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="comment">// Tabulate the probability of getting between zero heads and 0 upto 10 heads. |
| </span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting upto (<=) heads"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <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"><=</span> <span class="identifier">flips</span><span class="special">;</span> <span class="identifier">successes</span><span class="special">++)</span> |
| <span class="special">{</span> <span class="comment">// Say success means getting a head |
| </span> <span class="comment">// (equally success could mean getting a tail). |
| </span> <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 <= heads) |
| </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">successes</span> <span class="special"><<</span> <span class="string">" "</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">left</span> |
| <span class="special"><<</span> <span class="identifier">probability</span> <span class="special"><<</span> <span class="string">" or 1 in "</span> <span class="special"><<</span> <span class="number">1.</span> <span class="special">/</span> <span class="identifier">probability</span> <span class="special"><<</span> <span class="string">", or "</span> |
| <span class="special"><<</span> <span class="identifier">probability</span> <span class="special">*</span> <span class="number">100.</span> <span class="special"><<</span> <span class="string">"%"</span><span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> <span class="comment">// for i</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| The last (0 to 10 heads) must, of course, be 100% probability. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="special">}</span> |
| <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">&</span> <span class="identifier">e</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="comment">//</span></pre> |
| <p> |
| </p> |
| <p> |
| <a name="coinflip_eg_catch"></a> |
| </p> |
| <p> |
| It is always essential to include try & catch blocks because |
| default policies are to throw exceptions on arguments that are out |
| of domain or cause errors like numeric-overflow. |
| </p> |
| <p> |
| </p> |
| <p> |
| Lacking try & catch blocks, the program will abort, whereas the |
| message below from the thrown exception will give some helpful clues |
| as to the cause of the problem. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> |
| <span class="string">"\n"</span><span class="string">"Message from thrown exception was:\n "</span> <span class="special"><<</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| See <a href="../../../../../../../../example/binomial_coinflip_example.cpp" target="_top">binomial_coinflip_example.cpp</a> |
| for full source code, the program output looks like this: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Using Binomial distribution to predict how many heads and tails. |
| From 10 one can expect to get on average 5 heads (or tails). |
| Mode is 5 |
| Standard deviation is 1.581 |
| So about 2/3 will lie within 1 standard deviation and get between 4 and 6 correct. |
| Skewness is 0 |
| Skewness if success_fraction is 0.5 is 0 |
| |
| For 10 coin flips: |
| Probability of getting no heads is 0.0009766 |
| Probability of getting at least one head is 0.999 |
| Probability of getting 0 or 1 heads is 0.01074 |
| Probability of getting 0 or 1 (<= 1) heads is 0.01074 |
| Probability of getting 9 or 10 heads is 0.01074 |
| Probability of getting 9 or 10 heads is 0.01074 |
| Probability of getting 9 or 10 heads is 0.01074 |
| Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6562 |
| Probability of between 4 and 6 heads (4 or 5 or 6) is 0.6563 |
| Probability of between 3 and 7 heads (3, 4, 5, 6 or 7) is 0.8906 |
| |
| Probability of getting exactly (<code class="literal">=) heads |
| 0 0.0009766 or 1 in 1024, or 0.09766% |
| 1 0.009766 or 1 in 102.4, or 0.9766% |
| 2 0.04395 or 1 in 22.76, or 4.395% |
| 3 0.1172 or 1 in 8.533, or 11.72% |
| 4 0.2051 or 1 in 4.876, or 20.51% |
| 5 0.2461 or 1 in 4.063, or 24.61% |
| 6 0.2051 or 1 in 4.876, or 20.51% |
| 7 0.1172 or 1 in 8.533, or 11.72% |
| 8 0.04395 or 1 in 22.76, or 4.395% |
| 9 0.009766 or 1 in 102.4, or 0.9766% |
| 10 0.0009766 or 1 in 1024, or 0.09766% |
| |
| Probability of getting upto (<</code>) heads |
| 0 0.0009766 or 1 in 1024, or 0.09766% |
| 1 0.01074 or 1 in 93.09, or 1.074% |
| 2 0.05469 or 1 in 18.29, or 5.469% |
| 3 0.1719 or 1 in 5.818, or 17.19% |
| 4 0.377 or 1 in 2.653, or 37.7% |
| 5 0.623 or 1 in 1.605, or 62.3% |
| 6 0.8281 or 1 in 1.208, or 82.81% |
| 7 0.9453 or 1 in 1.058, or 94.53% |
| 8 0.9893 or 1 in 1.011, or 98.93% |
| 9 0.999 or 1 in 1.001, or 99.9% |
| 10 1 or 1 in 1, or 100% |
| </pre> |
| <p> |
| </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, 2010 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> |
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