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buffon_needle.html
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<h1 id="buffon-needle">buffon Needle</h1>
<p>habib ezatabadi (Stats9)</p>
<h1 id="code1-with-tidyverse-package">Code1 with tidyverse package:</h1>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" tabindex="-1"></a><span class="do">#####################################</span></span>
<span id="cb1-2"><a href="#cb1-2" tabindex="-1"></a></span>
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<span id="cb1-5"><a href="#cb1-5" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
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<span id="cb1-8"><a href="#cb1-8" tabindex="-1"></a>Buffon_needle <span class="ot"><-</span> <span class="cf">function</span>(<span class="at">n =</span> <span class="dv">25</span>, <span class="at">a =</span> <span class="fl">1.5</span>, <span class="at">L =</span> <span class="fl">1.5</span>, <span class="at">seed =</span> <span class="dv">1234</span>){</span>
<span id="cb1-9"><a href="#cb1-9" tabindex="-1"></a>Yintercept <span class="ot"><-</span> <span class="fu">seq</span>(<span class="sc">-</span><span class="dv">4</span>, <span class="dv">4</span>, <span class="at">by =</span> L)</span>
<span id="cb1-10"><a href="#cb1-10" tabindex="-1"></a></span>
<span id="cb1-11"><a href="#cb1-11" tabindex="-1"></a>P <span class="ot"><-</span> <span class="fu">ggplot</span>() <span class="sc">+</span></span>
<span id="cb1-12"><a href="#cb1-12" tabindex="-1"></a> <span class="fu">geom_hline</span>(<span class="at">yintercept =</span> Yintercept) <span class="sc">+</span> </span>
<span id="cb1-13"><a href="#cb1-13" tabindex="-1"></a> <span class="fu">labs</span>(<span class="at">y =</span> <span class="cn">NULL</span>) <span class="sc">+</span></span>
<span id="cb1-14"><a href="#cb1-14" tabindex="-1"></a> <span class="fu">coord_cartesian</span>(<span class="at">xlim =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">8</span>), <span class="at">ylim =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">4</span>, <span class="dv">4</span>), <span class="at">expand =</span> <span class="cn">TRUE</span>)<span class="sc">+</span></span>
<span id="cb1-15"><a href="#cb1-15" tabindex="-1"></a> <span class="fu">theme</span>(</span>
<span id="cb1-16"><a href="#cb1-16" tabindex="-1"></a> <span class="at">panel.border =</span> <span class="fu">element_blank</span>(),</span>
<span id="cb1-17"><a href="#cb1-17" tabindex="-1"></a> <span class="at">axis.text =</span> <span class="fu">element_blank</span>(),</span>
<span id="cb1-18"><a href="#cb1-18" tabindex="-1"></a> <span class="at">panel.grid =</span> <span class="fu">element_blank</span>(),</span>
<span id="cb1-19"><a href="#cb1-19" tabindex="-1"></a> <span class="at">axis.ticks =</span> <span class="fu">element_blank</span>(), </span>
<span id="cb1-20"><a href="#cb1-20" tabindex="-1"></a> <span class="at">panel.background =</span> <span class="fu">element_rect</span>(<span class="at">fill =</span> <span class="st">"white"</span>)</span>
<span id="cb1-21"><a href="#cb1-21" tabindex="-1"></a> )</span>
<span id="cb1-22"><a href="#cb1-22" tabindex="-1"></a>P</span>
<span id="cb1-23"><a href="#cb1-23" tabindex="-1"></a><span class="fu">set.seed</span>(seed)</span>
<span id="cb1-24"><a href="#cb1-24" tabindex="-1"></a>x1 <span class="ot"><-</span> <span class="fu">runif</span>(n, <span class="dv">1</span>, <span class="dv">7</span>)</span>
<span id="cb1-25"><a href="#cb1-25" tabindex="-1"></a>y1 <span class="ot"><-</span> <span class="fu">runif</span>(n, <span class="sc">-</span><span class="fl">3.5</span>, <span class="fl">3.5</span>)</span>
<span id="cb1-26"><a href="#cb1-26" tabindex="-1"></a>theta1 <span class="ot"><-</span> <span class="fu">runif</span>(n, <span class="dv">0</span>, pi)</span>
<span id="cb1-27"><a href="#cb1-27" tabindex="-1"></a>x1_2 <span class="ot"><-</span> x1 <span class="sc">+</span> a <span class="sc">*</span> <span class="fu">cos</span>(theta1)</span>
<span id="cb1-28"><a href="#cb1-28" tabindex="-1"></a>y1_2 <span class="ot"><-</span> y1 <span class="sc">+</span> a <span class="sc">*</span> <span class="fu">sin</span>(theta1)</span>
<span id="cb1-29"><a href="#cb1-29" tabindex="-1"></a></span>
<span id="cb1-30"><a href="#cb1-30" tabindex="-1"></a>X <span class="ot"><-</span> <span class="fu">numeric</span>(<span class="dv">2</span><span class="sc">*</span>n)</span>
<span id="cb1-31"><a href="#cb1-31" tabindex="-1"></a>Y <span class="ot"><-</span> <span class="fu">numeric</span>(<span class="dv">2</span><span class="sc">*</span>n)</span>
<span id="cb1-32"><a href="#cb1-32" tabindex="-1"></a>ind1 <span class="ot"><-</span> <span class="fu">seq</span>(<span class="dv">1</span>, <span class="dv">2</span><span class="sc">*</span>n, <span class="at">by =</span> <span class="dv">2</span>)</span>
<span id="cb1-33"><a href="#cb1-33" tabindex="-1"></a>ind2 <span class="ot"><-</span> <span class="fu">seq</span>(<span class="dv">2</span>, <span class="dv">2</span><span class="sc">*</span>n, <span class="at">by =</span> <span class="dv">2</span>)</span>
<span id="cb1-34"><a href="#cb1-34" tabindex="-1"></a>X[ind1] <span class="ot"><-</span> x1</span>
<span id="cb1-35"><a href="#cb1-35" tabindex="-1"></a>X[ind2] <span class="ot"><-</span> x1_2</span>
<span id="cb1-36"><a href="#cb1-36" tabindex="-1"></a>Y[ind1] <span class="ot"><-</span> y1</span>
<span id="cb1-37"><a href="#cb1-37" tabindex="-1"></a>Y[ind2] <span class="ot"><-</span> y1_2</span>
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<span id="cb1-39"><a href="#cb1-39" tabindex="-1"></a>Meanx <span class="ot"><-</span> (x1 <span class="sc">+</span> x1_2)<span class="sc">/</span><span class="dv">2</span></span>
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<span id="cb1-41"><a href="#cb1-41" tabindex="-1"></a>Meany <span class="ot"><-</span> (y1 <span class="sc">+</span> y1_2)<span class="sc">/</span><span class="dv">2</span></span>
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<span id="cb1-44"><a href="#cb1-44" tabindex="-1"></a>yy <span class="ot"><-</span> Yintercept</span>
<span id="cb1-45"><a href="#cb1-45" tabindex="-1"></a>nn <span class="ot"><-</span> <span class="fu">length</span>(Yintercept)</span>
<span id="cb1-46"><a href="#cb1-46" tabindex="-1"></a>Res <span class="ot"><-</span> <span class="fu">expand_grid</span>(Meany, yy) </span>
<span id="cb1-47"><a href="#cb1-47" tabindex="-1"></a>ID <span class="ot"><-</span> <span class="fu">rep</span>(<span class="fu">paste0</span>(<span class="st">"needle"</span>, <span class="dv">1</span><span class="sc">:</span>n), <span class="at">each =</span> nn)</span>
<span id="cb1-48"><a href="#cb1-48" tabindex="-1"></a>Res<span class="sc">$</span>ID <span class="ot">=</span> <span class="fu">factor</span>(ID, <span class="at">levels =</span> <span class="fu">unique</span>(ID), <span class="at">ordered =</span> <span class="cn">TRUE</span>)</span>
<span id="cb1-49"><a href="#cb1-49" tabindex="-1"></a>Res <span class="sc">%>%</span></span>
<span id="cb1-50"><a href="#cb1-50" tabindex="-1"></a> <span class="fu">mutate</span>(<span class="at">Distance =</span> <span class="fu">sqrt</span>((Meany <span class="sc">-</span> yy)<span class="sc">^</span><span class="dv">2</span>)) <span class="sc">%>%</span></span>
<span id="cb1-51"><a href="#cb1-51" tabindex="-1"></a> <span class="fu">group_by</span>(ID) <span class="sc">%>%</span> </span>
<span id="cb1-52"><a href="#cb1-52" tabindex="-1"></a> <span class="fu">summarise</span>(<span class="at">MIN =</span> <span class="fu">min</span>(Distance)) <span class="ot">-></span> DISTANCE</span>
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<span id="cb1-55"><a href="#cb1-55" tabindex="-1"></a>DISTANCE</span>
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<span id="cb1-57"><a href="#cb1-57" tabindex="-1"></a>needle_dist <span class="ot"><-</span> a <span class="sc">*</span> <span class="fu">sin</span>(theta1)<span class="sc">/</span><span class="dv">2</span></span>
<span id="cb1-58"><a href="#cb1-58" tabindex="-1"></a>DISTANCE <span class="ot"><-</span> DISTANCE <span class="sc">%>%</span></span>
<span id="cb1-59"><a href="#cb1-59" tabindex="-1"></a> <span class="fu">mutate</span>(<span class="at">HIT =</span> <span class="dv">1</span> <span class="sc">*</span> (MIN <span class="sc"><</span> needle_dist))</span>
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<span id="cb1-62"><a href="#cb1-62" tabindex="-1"></a></span>
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<span id="cb1-66"><a href="#cb1-66" tabindex="-1"></a>IND_hit <span class="ot"><-</span> <span class="fu">which</span>(DISTANCE<span class="sc">$</span>HIT <span class="sc">==</span> <span class="dv">1</span>)</span>
<span id="cb1-67"><a href="#cb1-67" tabindex="-1"></a>n_hit <span class="ot"><-</span> <span class="fu">length</span>(IND_hit)</span>
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<span id="cb1-69"><a href="#cb1-69" tabindex="-1"></a>df2 <span class="ot"><-</span> <span class="fu">tibble</span>(<span class="at">x =</span> Meanx[IND_hit], <span class="at">y =</span> Meany[IND_hit])</span>
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<span id="cb1-72"><a href="#cb1-72" tabindex="-1"></a>Df <span class="ot"><-</span> <span class="fu">tibble</span>(<span class="at">x =</span> X, <span class="at">y =</span> Y, <span class="at">Time =</span> <span class="fu">rep</span>(<span class="dv">1</span><span class="sc">:</span>n, <span class="at">each =</span> <span class="dv">2</span>))</span>
<span id="cb1-73"><a href="#cb1-73" tabindex="-1"></a>P <span class="sc">+</span> <span class="fu">geom_line</span>(<span class="at">data =</span> Df, <span class="fu">aes</span>(<span class="at">x =</span> x, <span class="at">y =</span> y, <span class="at">group =</span> Time), </span>
<span id="cb1-74"><a href="#cb1-74" tabindex="-1"></a> <span class="at">linewidth =</span> <span class="fl">1.2</span>, <span class="at">color =</span> <span class="st">"darkblue"</span>, <span class="at">alpha =</span> .<span class="dv">7</span>) <span class="sc">+</span> </span>
<span id="cb1-75"><a href="#cb1-75" tabindex="-1"></a> <span class="fu">geom_point</span>(<span class="at">data =</span> df2, <span class="fu">aes</span>(<span class="at">x =</span> x, <span class="at">y =</span> y), <span class="at">shape =</span> <span class="dv">4</span>, </span>
<span id="cb1-76"><a href="#cb1-76" tabindex="-1"></a> <span class="at">color =</span> <span class="st">"orange"</span>, <span class="at">stroke =</span> <span class="dv">3</span>, <span class="at">size =</span> <span class="dv">3</span>) <span class="sc">+</span> </span>
<span id="cb1-77"><a href="#cb1-77" tabindex="-1"></a> <span class="fu">labs</span>(<span class="at">title =</span> <span class="fu">paste0</span>(<span class="st">"# Hits = "</span>, n_hit), <span class="at">x =</span> <span class="cn">NULL</span>) <span class="ot">-></span> P1</span>
<span id="cb1-78"><a href="#cb1-78" tabindex="-1"></a>myresult <span class="ot"><-</span> <span class="fu">list</span>(<span class="at">number_hits =</span> n_hit, <span class="at">Plot =</span> P1)</span>
<span id="cb1-79"><a href="#cb1-79" tabindex="-1"></a><span class="fu">return</span>(myresult)</span>
<span id="cb1-80"><a href="#cb1-80" tabindex="-1"></a>}</span>
<span id="cb1-81"><a href="#cb1-81" tabindex="-1"></a></span>
<span id="cb1-82"><a href="#cb1-82" tabindex="-1"></a>outPut <span class="ot"><-</span> <span class="fu">Buffon_needle</span>()</span>
<span id="cb1-83"><a href="#cb1-83" tabindex="-1"></a>P1 <span class="ot"><-</span> outPut<span class="sc">$</span>Plot</span>
<span id="cb1-84"><a href="#cb1-84" tabindex="-1"></a>P1</span></code></pre></div>
<p><img src="data:image/png;base64,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" /><!-- --></p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" tabindex="-1"></a><span class="do">## if you want show this plot as a giff then run these codes below</span></span>
<span id="cb2-2"><a href="#cb2-2" tabindex="-1"></a></span>
<span id="cb2-3"><a href="#cb2-3" tabindex="-1"></a><span class="co">#if (! require(gganimate)) {</span></span>
<span id="cb2-4"><a href="#cb2-4" tabindex="-1"></a><span class="co"># install.packages("gganimate")</span></span>
<span id="cb2-5"><a href="#cb2-5" tabindex="-1"></a><span class="co"># library(gganimate)</span></span>
<span id="cb2-6"><a href="#cb2-6" tabindex="-1"></a><span class="co">#}</span></span>
<span id="cb2-7"><a href="#cb2-7" tabindex="-1"></a></span>
<span id="cb2-8"><a href="#cb2-8" tabindex="-1"></a><span class="co"># P1 + transition_reveal(Time)</span></span></code></pre></div>
<hr />
<h1 id="code2-with-base-functions-in-r">code2: with base functions in
R</h1>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" tabindex="-1"></a>Buffon_updated <span class="ot"><-</span> <span class="cf">function</span>(<span class="at">n =</span> <span class="dv">20</span>, <span class="at">a =</span> <span class="fl">1.5</span>, <span class="at">L =</span> <span class="dv">2</span>, </span>
<span id="cb3-2"><a href="#cb3-2" tabindex="-1"></a> <span class="at">show =</span> <span class="cn">TRUE</span>, <span class="at">seed =</span> <span class="dv">12</span>){</span>
<span id="cb3-3"><a href="#cb3-3" tabindex="-1"></a><span class="cf">if</span>(a <span class="sc">></span> L) <span class="fu">stop</span>(<span class="st">"a must be smaller than L"</span>)</span>
<span id="cb3-4"><a href="#cb3-4" tabindex="-1"></a></span>
<span id="cb3-5"><a href="#cb3-5" tabindex="-1"></a>Yintercept <span class="ot"><-</span> <span class="fu">seq</span>(<span class="sc">-</span><span class="dv">4</span>, <span class="dv">4</span>, <span class="at">by =</span> L)</span>
<span id="cb3-6"><a href="#cb3-6" tabindex="-1"></a></span>
<span id="cb3-7"><a href="#cb3-7" tabindex="-1"></a><span class="fu">set.seed</span>(seed)</span>
<span id="cb3-8"><a href="#cb3-8" tabindex="-1"></a>x1 <span class="ot"><-</span> <span class="fu">runif</span>(n, <span class="dv">1</span>, <span class="dv">7</span>)</span>
<span id="cb3-9"><a href="#cb3-9" tabindex="-1"></a>y1 <span class="ot"><-</span> <span class="fu">runif</span>(n, <span class="sc">-</span><span class="fl">3.5</span>, <span class="fl">3.5</span>)</span>
<span id="cb3-10"><a href="#cb3-10" tabindex="-1"></a>theta1 <span class="ot"><-</span> <span class="fu">runif</span>(n, <span class="dv">0</span>, pi)</span>
<span id="cb3-11"><a href="#cb3-11" tabindex="-1"></a>x1_2 <span class="ot"><-</span> x1 <span class="sc">+</span> a <span class="sc">*</span> <span class="fu">cos</span>(theta1)</span>
<span id="cb3-12"><a href="#cb3-12" tabindex="-1"></a>y1_2 <span class="ot"><-</span> y1 <span class="sc">+</span> a <span class="sc">*</span> <span class="fu">sin</span>(theta1)</span>
<span id="cb3-13"><a href="#cb3-13" tabindex="-1"></a></span>
<span id="cb3-14"><a href="#cb3-14" tabindex="-1"></a></span>
<span id="cb3-15"><a href="#cb3-15" tabindex="-1"></a></span>
<span id="cb3-16"><a href="#cb3-16" tabindex="-1"></a></span>
<span id="cb3-17"><a href="#cb3-17" tabindex="-1"></a>Meanx <span class="ot"><-</span> (x1 <span class="sc">+</span> x1_2)<span class="sc">/</span><span class="dv">2</span></span>
<span id="cb3-18"><a href="#cb3-18" tabindex="-1"></a></span>
<span id="cb3-19"><a href="#cb3-19" tabindex="-1"></a>Meany <span class="ot"><-</span> (y1 <span class="sc">+</span> y1_2)<span class="sc">/</span><span class="dv">2</span></span>
<span id="cb3-20"><a href="#cb3-20" tabindex="-1"></a></span>
<span id="cb3-21"><a href="#cb3-21" tabindex="-1"></a></span>
<span id="cb3-22"><a href="#cb3-22" tabindex="-1"></a></span>
<span id="cb3-23"><a href="#cb3-23" tabindex="-1"></a></span>
<span id="cb3-24"><a href="#cb3-24" tabindex="-1"></a></span>
<span id="cb3-25"><a href="#cb3-25" tabindex="-1"></a></span>
<span id="cb3-26"><a href="#cb3-26" tabindex="-1"></a></span>
<span id="cb3-27"><a href="#cb3-27" tabindex="-1"></a></span>
<span id="cb3-28"><a href="#cb3-28" tabindex="-1"></a></span>
<span id="cb3-29"><a href="#cb3-29" tabindex="-1"></a>nn <span class="ot"><-</span> <span class="fu">length</span>(Yintercept)</span>
<span id="cb3-30"><a href="#cb3-30" tabindex="-1"></a>Res <span class="ot"><-</span> <span class="fu">expand.grid</span>(Yintercept, Meany) </span>
<span id="cb3-31"><a href="#cb3-31" tabindex="-1"></a><span class="fu">names</span>(Res) <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"Meany"</span>, <span class="st">"y"</span>)</span>
<span id="cb3-32"><a href="#cb3-32" tabindex="-1"></a>ID <span class="ot"><-</span> <span class="fu">rep</span>(<span class="fu">paste0</span>(<span class="st">"needle"</span>, <span class="dv">1</span><span class="sc">:</span>n), <span class="at">each =</span> nn)</span>
<span id="cb3-33"><a href="#cb3-33" tabindex="-1"></a>Res<span class="sc">$</span>ID <span class="ot">=</span> <span class="fu">factor</span>(ID, <span class="at">levels =</span> <span class="fu">unique</span>(ID), <span class="at">ordered =</span> <span class="cn">TRUE</span>)</span>
<span id="cb3-34"><a href="#cb3-34" tabindex="-1"></a>Distance <span class="ot">=</span> <span class="fu">sqrt</span>((Res<span class="sc">$</span>Meany <span class="sc">-</span> Res<span class="sc">$</span>y)<span class="sc">^</span><span class="dv">2</span>)</span>
<span id="cb3-35"><a href="#cb3-35" tabindex="-1"></a>Res<span class="sc">$</span>distance <span class="ot"><-</span> Distance</span>
<span id="cb3-36"><a href="#cb3-36" tabindex="-1"></a>result <span class="ot"><-</span> <span class="fu">aggregate</span>(Distance <span class="sc">~</span> ID, <span class="at">data =</span> Res, <span class="at">FUN =</span> min)</span>
<span id="cb3-37"><a href="#cb3-37" tabindex="-1"></a></span>
<span id="cb3-38"><a href="#cb3-38" tabindex="-1"></a></span>
<span id="cb3-39"><a href="#cb3-39" tabindex="-1"></a>needle_dist <span class="ot"><-</span> a <span class="sc">*</span> <span class="fu">sin</span>(theta1)<span class="sc">/</span><span class="dv">2</span></span>
<span id="cb3-40"><a href="#cb3-40" tabindex="-1"></a></span>
<span id="cb3-41"><a href="#cb3-41" tabindex="-1"></a>Hit <span class="ot"><-</span> <span class="dv">1</span> <span class="sc">*</span> (result<span class="sc">$</span>Distance <span class="sc"><</span> needle_dist)</span>
<span id="cb3-42"><a href="#cb3-42" tabindex="-1"></a></span>
<span id="cb3-43"><a href="#cb3-43" tabindex="-1"></a></span>
<span id="cb3-44"><a href="#cb3-44" tabindex="-1"></a></span>
<span id="cb3-45"><a href="#cb3-45" tabindex="-1"></a></span>
<span id="cb3-46"><a href="#cb3-46" tabindex="-1"></a></span>
<span id="cb3-47"><a href="#cb3-47" tabindex="-1"></a></span>
<span id="cb3-48"><a href="#cb3-48" tabindex="-1"></a></span>
<span id="cb3-49"><a href="#cb3-49" tabindex="-1"></a>IND_hit <span class="ot"><-</span> <span class="fu">which</span>(Hit <span class="sc">==</span> <span class="dv">1</span>)</span>
<span id="cb3-50"><a href="#cb3-50" tabindex="-1"></a>n_hit <span class="ot"><-</span> <span class="fu">length</span>(IND_hit)</span>
<span id="cb3-51"><a href="#cb3-51" tabindex="-1"></a>df2 <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="at">x =</span> Meanx[IND_hit], <span class="at">y =</span> Meany[IND_hit])</span>
<span id="cb3-52"><a href="#cb3-52" tabindex="-1"></a></span>
<span id="cb3-53"><a href="#cb3-53" tabindex="-1"></a></span>
<span id="cb3-54"><a href="#cb3-54" tabindex="-1"></a></span>
<span id="cb3-55"><a href="#cb3-55" tabindex="-1"></a></span>
<span id="cb3-56"><a href="#cb3-56" tabindex="-1"></a><span class="cf">if</span>(show){</span>
<span id="cb3-57"><a href="#cb3-57" tabindex="-1"></a><span class="fu">plot</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">8</span>), <span class="at">y =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">4</span>, <span class="dv">4</span>), <span class="at">type =</span> <span class="st">"n"</span>, <span class="at">frame =</span> <span class="cn">FALSE</span>, </span>
<span id="cb3-58"><a href="#cb3-58" tabindex="-1"></a><span class="at">xaxt =</span> <span class="st">"n"</span>, <span class="at">yaxt =</span> <span class="st">"n"</span>, <span class="at">xlab =</span> <span class="st">""</span>, <span class="at">ylab =</span> <span class="st">""</span>, </span>
<span id="cb3-59"><a href="#cb3-59" tabindex="-1"></a><span class="at">main =</span> <span class="fu">paste0</span>(<span class="st">"# Hits = "</span>, n_hit))</span>
<span id="cb3-60"><a href="#cb3-60" tabindex="-1"></a></span>
<span id="cb3-61"><a href="#cb3-61" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">h =</span> Yintercept)</span>
<span id="cb3-62"><a href="#cb3-62" tabindex="-1"></a><span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>n){</span>
<span id="cb3-63"><a href="#cb3-63" tabindex="-1"></a>xx <span class="ot"><-</span> <span class="fu">c</span>(x1[i], x1_2[i])</span>
<span id="cb3-64"><a href="#cb3-64" tabindex="-1"></a>yy <span class="ot"><-</span> <span class="fu">c</span>(y1[i], y1_2[i])</span>
<span id="cb3-65"><a href="#cb3-65" tabindex="-1"></a><span class="fu">lines</span>(<span class="at">x =</span> xx, <span class="at">y =</span> yy, <span class="at">col =</span> <span class="st">"blue"</span>, <span class="at">lwd =</span> <span class="dv">2</span>)</span>
<span id="cb3-66"><a href="#cb3-66" tabindex="-1"></a>}</span>
<span id="cb3-67"><a href="#cb3-67" tabindex="-1"></a></span>
<span id="cb3-68"><a href="#cb3-68" tabindex="-1"></a><span class="fu">points</span>(<span class="at">x =</span> df2<span class="sc">$</span>x, <span class="at">y =</span> df2<span class="sc">$</span>y, <span class="at">pch =</span> <span class="dv">4</span>, </span>
<span id="cb3-69"><a href="#cb3-69" tabindex="-1"></a> <span class="at">col =</span> <span class="st">"tomato"</span>, <span class="at">lwd =</span> <span class="dv">3</span>)</span>
<span id="cb3-70"><a href="#cb3-70" tabindex="-1"></a></span>
<span id="cb3-71"><a href="#cb3-71" tabindex="-1"></a>}</span>
<span id="cb3-72"><a href="#cb3-72" tabindex="-1"></a><span class="fu">return</span>(n_hit)</span>
<span id="cb3-73"><a href="#cb3-73" tabindex="-1"></a>}</span>
<span id="cb3-74"><a href="#cb3-74" tabindex="-1"></a></span>
<span id="cb3-75"><a href="#cb3-75" tabindex="-1"></a></span>
<span id="cb3-76"><a href="#cb3-76" tabindex="-1"></a><span class="fu">Buffon_updated</span>(<span class="at">L =</span> <span class="dv">1</span>, <span class="at">a =</span> <span class="dv">1</span>, <span class="at">seed =</span> <span class="dv">2</span>, <span class="at">n =</span> <span class="dv">20</span>)</span></code></pre></div>
<p><img src="data:image/png;base64,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" /><!-- --></p>
<pre><code>## [1] 11</code></pre>
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