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<li><a href="#introduction" id="toc-introduction" class="nav-link active" data-scroll-target="#introduction"><span class="header-section-number">10.1</span> Introduction</a></li>
<li><a href="#sec-slice-indexing" id="toc-sec-slice-indexing" class="nav-link" data-scroll-target="#sec-slice-indexing"><span class="header-section-number">10.2</span> Selecting elements from vectors with slicing</a></li>
<li><a href="#the-treasure-fleet-recovered" id="toc-the-treasure-fleet-recovered" class="nav-link" data-scroll-target="#the-treasure-fleet-recovered"><span class="header-section-number">10.3</span> The treasure fleet recovered</a></li>
<li><a href="#back-to-boolean-s" id="toc-back-to-boolean-s" class="nav-link" data-scroll-target="#back-to-boolean-s"><span class="header-section-number">10.4</span> Back to Boolean vectors</a></li>
<li><a href="#sec-ships-booleans" id="toc-sec-ships-booleans" class="nav-link" data-scroll-target="#sec-ships-booleans"><span class="header-section-number">10.5</span> Boolean vectors and another take on the ships problem</a></li>
<li><a href="#sec-combine-booleans" id="toc-sec-combine-booleans" class="nav-link" data-scroll-target="#sec-combine-booleans">10.6 Combining Boolean arrays</a></li>
<li><a href="#the-monty-hall-problem" id="toc-the-monty-hall-problem" class="nav-link" data-scroll-target="#the-monty-hall-problem"><span class="header-section-number">10.7</span> The Monty Hall problem</a></li>
<li><a href="#monty-hall-with" id="toc-monty-hall-with" class="nav-link" data-scroll-target="#monty-hall-with"><span class="header-section-number">10.8</span> Monty Hall with R</a>
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<li><a href="#insight-from-the-monty-hall-simulation" id="toc-insight-from-the-monty-hall-simulation" class="nav-link" data-scroll-target="#insight-from-the-monty-hall-simulation">10.8.1 Insight from the Monty Hall simulation</a></li>
<li><a href="#simulation-and-a-variant-of-monty-hall" id="toc-simulation-and-a-variant-of-monty-hall" class="nav-link" data-scroll-target="#simulation-and-a-variant-of-monty-hall">10.8.2 Simulation and a variant of Monty Hall</a></li>
</ul></li>
<li><a href="#why-use-simulation" id="toc-why-use-simulation" class="nav-link" data-scroll-target="#why-use-simulation"><span class="header-section-number">10.9</span> Why use simulation?</a></li>
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<h1 class="title"><span id="sec-more-sampling-tools" class="quarto-section-identifier"><span class="chapter-number">10</span> <span class="chapter-title">Two puzzles and more tools</span></span></h1>
</div>
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</header>
<section id="introduction" class="level2" data-number="10.1">
<h2 data-number="10.1" class="anchored" data-anchor-id="introduction"><span class="header-section-number">10.1</span> Introduction</h2>
<p>In the next chapter we will deal with some more involved problems in <em>probability</em>, as a preparation for <em>statistics</em>, where we use reasoning from probability to draw conclusions about a world like our own, where variation often appears to be more or less random.</p>
<p>Before we get down to the business of complex probabilistic problems in the next few chapters, let’s consider a couple of peculiar puzzles. These puzzles allow us to introduce some more of the key tools in R for Monte Carlo resampling, and show the power of such simulation to help solve, and then reason about, problems in probability.</p>
<p>But, before we get to the puzzles, we will need some more elements of R.</p>
</section>
<section id="sec-slice-indexing" class="level2" data-number="10.2">
<h2 data-number="10.2" class="anchored" data-anchor-id="sec-slice-indexing"><span class="header-section-number">10.2</span> Selecting elements from vectors with slicing</h2>
<div id="nte-selecting_by_slicing" class="callout callout-style-default callout-note callout-titled">
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Note 10.1: Notebook: Selecting elements by slicing
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<div class="callout-body-container callout-body">
<div class="nb-links">
<p><a class="notebook-link" href="notebooks/selecting_by_slicing.Rmd">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=selecting_by_slicing.ipynb">Interact</a></p>
</div>
</div>
</div>
<div class="nb-start" name="selecting_by_slicing" title="Selecting elements by slicing">
</div>
<p>As you saw in <a href="resampling_with_code2.html#sec-array-indexing" class="quarto-xref"><span>Section 6.6</span></a>, we do <em>indexing</em> when we put square brackets following a value that is a container, such as a vector. Inside the square brackets we put another value to specify which elements we want to fetch from the container.</p>
<p>We will use the <code>some_numbers</code> vector as our container for indexing:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>some_numbers <span class="ot">=</span> <span class="fu">c</span>(<span class="dv">3</span>, <span class="dv">1</span>, <span class="dv">4</span>, <span class="dv">1</span>, <span class="dv">5</span>, <span class="dv">9</span>, <span class="dv">2</span>, <span class="dv">6</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>In the indexing expression below, we have the vector <code>some_numbers</code>, followed by the value <code>3</code> inside the square brackets, telling R that we want the third value from the <code>some_numbers</code> container.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Indexing with an integer inside the square brackets.</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>some_numbers[<span class="dv">3</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 4</code></pre>
</div>
</div>
<p>In the example above, we put an integer (a whole number) inside the square brackets, to specify the position of the element we want to fetch from the container.</p>
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<div class="r">
<p>We can also put a <em>vector</em> inside the square brackets. If we put a vector inside the square brackets, the elements of that vector specify the positions of the elements we want to fetch.</p>
<p>We have already seen that R can form a vector of sequential integers by using the colon (<code>:</code>) operator.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The sequence of integers from 3 through 8</span></span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span><span class="sc">:</span><span class="dv">8</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 3 4 5 6 7 8</code></pre>
</div>
</div>
<p>See <a href="testing_counts_2.html#nte-what-is-an-operator" class="quarto-xref">Note <span>23.2</span></a> for more on operators. The colon operator (the <code>:</code>) serves as a signal to R to create a new value by <em>operating</em> on the values to the left and right of the colon.</p>
<p>The integer to the left of the colon specifies the <em>start</em> position; this is the position of the first element we want. The integer to the right of the colon gives the <em>stop</em> position. R interprets this as asking for a vector containing all the integers from (including) the <em>start</em> value (3 in the example above) through (including) the stop value (8 in the example above).</p>
<p>For example, here we index the vector <code>some_numbers</code> with another vector having <em>start</em> position of 2 and <em>stop</em> position of 5. The result is another vector, that has the elements of <code>some_numbers</code> from positions 2 through 5:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Indexing a vector with a vector.</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>our_positions <span class="ot"><-</span> <span class="dv">2</span><span class="sc">:</span><span class="dv">5</span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a>some_numbers[our_positions]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 1 4 1 5</code></pre>
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</div>
<p>Of course we could also do the same thing in one line, by forming the vector of positions inside the square (indexing) brackets:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Indexing a vector with a vector.</span></span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>some_numbers[<span class="dv">2</span><span class="sc">:</span><span class="dv">5</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 1 4 1 5</code></pre>
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End of notebook: Selecting elements by slicing
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<div class="callout-body-container callout-body">
<p><code>selecting_by_slicing</code> starts at <a href="#nte-selecting_by_slicing" class="quarto-xref">Note <span>10.1</span></a>.</p>
</div>
</div>
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</section>
<section id="the-treasure-fleet-recovered" class="level2" data-number="10.3">
<h2 data-number="10.3" class="anchored" data-anchor-id="the-treasure-fleet-recovered"><span class="header-section-number">10.3</span> The treasure fleet recovered</h2>
<p>As promised, we have now arrived at the first of the probability puzzles.</p>
<p>This is a classic problem in probability:<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a></p>
<blockquote class="blockquote">
<p>A Spanish treasure fleet of three ships was sunk at sea off Mexico. One ship had a chest of gold forward and another aft, another ship had a chest of gold forward and a chest of silver aft, while a third ship had a chest of silver forward and another chest of silver aft. Divers just found one of the ships and a chest of gold in it, but they don’t know whether it was from forward or aft. They are now taking bets about whether the other chest found on the same ship will contain silver or gold. What are fair odds?</p>
</blockquote>
<p>These are the logical steps one may distinguish in arriving at a correct answer with deductive logic (portrayed in <a href="#fig-ships-gold-silver" class="quarto-xref">Figure <span>10.1</span></a>).</p>
<ol type="1">
<li><p>Postulate three ships — Ship I with two gold chests (G-G), ship II with one gold and one silver chest (G-S), and ship III with S-S. (Choosing notation might well be considered one or more additional steps.)</p></li>
<li><p>Assert equal probabilities of each ship being found.</p></li>
<li><p>Step 2 implies equal probabilities of being found for each of the six chests.</p></li>
<li><p>Fact: Diver finds a chest of gold.</p></li>
<li><p>Step 4 implies that S-S ship III was not found; hence remove it from subsequent analysis.</p></li>
<li><p>Three possibilities: 6a) Diver found chest I-Ga, 6b) diver found I-Gb, 6c) diver found II-Gc.</p>
<p>From step 2, the cases a, b, and c in step 6 have equal probabilities.</p></li>
<li><p>If possibility 6a is the case, then the other chest is I-Gb; the comparable statements for cases 6b and 6c are I-Ga and II-S.</p></li>
<li><p>From steps 6 and 7: From equal probabilities of the three cases, and no other possible outcome, <span class="math inline">\(P(6a) = 1/3\)</span>, <span class="math inline">\(P(6b) = 1/3\)</span>, <span class="math inline">\(P(6c) =
1/3\)</span>.</p></li>
<li><p>So <span class="math inline">\(P(G) = P(6a) + P(6b)\)</span> = 1/3 + 1/3 = 2/3.</p></li>
</ol>
<p>See <a href="#fig-ships-gold-silver" class="quarto-xref">Figure <span>10.1</span></a>.</p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div id="fig-ships-gold-silver" class="quarto-float quarto-figure quarto-figure-center anchored" data-fig-align="center">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-ships-gold-silver-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="diagrams/ships_gold_silver.svg" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-ships-gold-silver-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 10.1: Ships with Gold and Silver
</figcaption>
</figure>
</div>
</div>
</div>
<p>The following simulation arrives at the correct answer.</p>
<ol type="1">
<li>Write “Gold” on three pieces of paper and “Silver” on three pieces of paper. These represent the chests.</li>
<li>Get three buckets each with two pieces of paper. Each bucket represents a ship, each piece of paper represents a chest in that ship. One bucket has two pieces of paper with “Gold” written on them; one has pieces of paper with “Gold” and “Silver”, and one has “Silver” and “Silver”.</li>
<li>Choose a bucket at random, to represent choosing a ship at random.</li>
<li>Shuffle the pieces of paper in the bucket and pick one, to represent choosing the first chest from that ship at random.</li>
<li>If the piece of paper says “Silver”, the first chest we found in this ship was silver, and we stop the trial and make no further record. If “Gold”, continue.</li>
<li>Get the second piece of paper from the bucket, representing the second chest on the chosen ship. Record whether this was “Silver” or “Gold” on the scoreboard.</li>
<li>Repeat steps (3 - 6) many times, and calculate the proportion of “Gold”s on the scoreboard. (The answer should be about <span class="math inline">\(\frac{2}{3}\)</span>.)</li>
</ol>
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<p>Here is a notebook simulation with R:</p>
<div id="nte-gold_silver_ships" class="callout callout-style-default callout-note callout-titled">
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Note 10.2: Notebook: Ships with gold and silver
</div>
</div>
<div class="callout-body-container callout-body">
<div class="nb-links">
<p><a class="notebook-link" href="notebooks/gold_silver_ships.Rmd">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=gold_silver_ships.ipynb">Interact</a></p>
</div>
</div>
</div>
<div class="nb-start" name="gold_silver_ships" title="Ships with gold and silver">
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The 3 buckets. Each bucket represents a ship. Each has two chests.</span></span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a>bucket1 <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">'Gold'</span>, <span class="st">'Gold'</span>) <span class="co"># Chests in first ship.</span></span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a>bucket2 <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">'Gold'</span>, <span class="st">'Silver'</span>) <span class="co"># Chests in second ship.</span></span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a>bucket3 <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">'Silver'</span>, <span class="st">'Silver'</span>) <span class="co"># Chests in third ship.</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Mark trials as not valid to start with.</span></span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Trials where we don't get a gold chest first will</span></span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a><span class="co"># keep this 'No gold in chest 1, chest 2 never opened' marker.</span></span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a>second_chests <span class="ot"><-</span> <span class="fu">rep</span>(<span class="st">'No gold in chest 1, chest 2 never opened'</span>, <span class="dv">10000</span>)</span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">10000</span>) {</span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a> <span class="co"># Select a ship at random from the three ships.</span></span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a> ship_no <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>, <span class="at">size=</span><span class="dv">1</span>)</span>
<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a> <span class="co"># Get the chests from this ship (represented by a bucket).</span></span>
<span id="cb11-10"><a href="#cb11-10" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (ship_no <span class="sc">==</span> <span class="dv">1</span>) {</span>
<span id="cb11-11"><a href="#cb11-11" aria-hidden="true" tabindex="-1"></a> bucket <span class="ot"><-</span> bucket1</span>
<span id="cb11-12"><a href="#cb11-12" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb11-13"><a href="#cb11-13" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (ship_no <span class="sc">==</span> <span class="dv">2</span>) {</span>
<span id="cb11-14"><a href="#cb11-14" aria-hidden="true" tabindex="-1"></a> bucket <span class="ot"><-</span> bucket2</span>
<span id="cb11-15"><a href="#cb11-15" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb11-16"><a href="#cb11-16" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (ship_no <span class="sc">==</span> <span class="dv">3</span>) {</span>
<span id="cb11-17"><a href="#cb11-17" aria-hidden="true" tabindex="-1"></a> bucket <span class="ot"><-</span> bucket3</span>
<span id="cb11-18"><a href="#cb11-18" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb11-19"><a href="#cb11-19" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-20"><a href="#cb11-20" aria-hidden="true" tabindex="-1"></a> <span class="co"># We shuffle the order of the chests in this ship, to simulate</span></span>
<span id="cb11-21"><a href="#cb11-21" aria-hidden="true" tabindex="-1"></a> <span class="co"># the fact that we don't know which of the two chests we have</span></span>
<span id="cb11-22"><a href="#cb11-22" aria-hidden="true" tabindex="-1"></a> <span class="co"># found first.</span></span>
<span id="cb11-23"><a href="#cb11-23" aria-hidden="true" tabindex="-1"></a> shuffled <span class="ot"><-</span> <span class="fu">sample</span>(bucket)</span>
<span id="cb11-24"><a href="#cb11-24" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-25"><a href="#cb11-25" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (shuffled[<span class="dv">1</span>] <span class="sc">==</span> <span class="st">'Gold'</span>) { <span class="co"># We found a gold chest first.</span></span>
<span id="cb11-26"><a href="#cb11-26" aria-hidden="true" tabindex="-1"></a> <span class="co"># Store whether the Second chest was silver or gold.</span></span>
<span id="cb11-27"><a href="#cb11-27" aria-hidden="true" tabindex="-1"></a> second_chests[i] <span class="ot"><-</span> shuffled[<span class="dv">2</span>]</span>
<span id="cb11-28"><a href="#cb11-28" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb11-29"><a href="#cb11-29" aria-hidden="true" tabindex="-1"></a>} <span class="co"># End loop, go back to beginning.</span></span>
<span id="cb11-30"><a href="#cb11-30" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-31"><a href="#cb11-31" aria-hidden="true" tabindex="-1"></a><span class="co"># Number of times we found gold in the second chest.</span></span>
<span id="cb11-32"><a href="#cb11-32" aria-hidden="true" tabindex="-1"></a>n_golds <span class="ot"><-</span> <span class="fu">sum</span>(second_chests <span class="sc">==</span> <span class="st">'Gold'</span>)</span>
<span id="cb11-33"><a href="#cb11-33" aria-hidden="true" tabindex="-1"></a><span class="co"># Number of times we found silver in the second chest.</span></span>
<span id="cb11-34"><a href="#cb11-34" aria-hidden="true" tabindex="-1"></a>n_silvers <span class="ot"><-</span> <span class="fu">sum</span>(second_chests <span class="sc">==</span> <span class="st">'Silver'</span>)</span>
<span id="cb11-35"><a href="#cb11-35" aria-hidden="true" tabindex="-1"></a><span class="co"># As a ratio of golds to all second chests (where the first was gold).</span></span>
<span id="cb11-36"><a href="#cb11-36" aria-hidden="true" tabindex="-1"></a><span class="fu">message</span>(n_golds <span class="sc">/</span> (n_golds <span class="sc">+</span> n_silvers))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stderr">
<pre><code>0.655882352941176</code></pre>
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End of notebook: Ships with gold and silver
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<p><code>gold_silver_ships</code> starts at <a href="#nte-gold_silver_ships" class="quarto-xref">Note <span>10.2</span></a>.</p>
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<p>In the code above, we have first chosen the ship number at random, and then used a set of <code>if ...</code> statements to get the pair of chests corresponding to the given ship. There are simpler and more elegant ways of writing this code, but they would need some R features that we haven’t covered yet.<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a></p>
</section>
<section id="back-to-boolean-s" class="level2" data-number="10.4">
<h2 data-number="10.4" class="anchored" data-anchor-id="back-to-boolean-s"><span class="header-section-number">10.4</span> Back to Boolean vectors</h2>
<p>The code above implements the procedure we might well use if we were simulating the problem physically. We do a trial, and we record the result. We do this on a piece of paper if we are doing a physical simulation, and in the <code>second_chests</code> vector in code.</p>
<p>Finally we tally up the results. If we are doing a physical simulation, we go back over the all the trial results and counting up the “Gold” and “Silver” outcomes. In code we use the comparisons <code>== 'Gold'</code> and <code>== 'Silver'</code> to find the trials of interest, and then count them up with <code>sum</code>.</p>
<p>Boolean (logical) vectors are a fundamental tool in R, and we will use them in nearly all our simulations.</p>
<p>Here is a remind of how those vectors work.</p>
<p>First, let’s slice out the first 10 values of the <code>second_chests</code> trial-by-trial results tally from the simulation above:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Get values at positions 1 through 10</span></span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a>first_10_chests <span class="ot"><-</span> second_chests[<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>]</span>
<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a>first_10_chests</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> [1] "Gold"
[2] "No gold in chest 1, chest 2 never opened"
[3] "No gold in chest 1, chest 2 never opened"
[4] "Silver"
[5] "Gold"
[6] "No gold in chest 1, chest 2 never opened"
[7] "Silver"
[8] "Silver"
[9] "Gold"
[10] "No gold in chest 1, chest 2 never opened"</code></pre>
</div>
</div>
<p>Before we started the simulation, we set <code>second_chests</code> to contain 10,000 strings, where each string was “No gold in chest 1, chest 2 never opened”. In the simulation, we check whether there was gold in the first chest, and, if not, we don’t change the value in <code>second_chest</code>, and the value remains as “No gold in chest 1, chest 2 never opened”.</p>
<p>Only if there was gold in the first chest, do we go on to check whether the second chest contains silver or gold. Therefore, we only set a new value in <code>second_chests</code> where there was gold in the first chest.</p>
<p>Now let’s show the effect of running a comparison on <code>first_10_chests</code>:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>were_gold <span class="ot"><-</span> (first_10_chests <span class="sc">==</span> <span class="st">'Gold'</span>)</span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a>were_gold</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code> [1] TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE TRUE FALSE</code></pre>
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Parentheses and Boolean comparisons
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<p>Notice the round brackets (parentheses) around <code>(first_10_chests == 'Gold')</code>. In this particular case, we would get the same result without the parentheses, so the paretheses are optional. In general, you will see we put parentheses around all expressions that generate Boolean vectors, and we recommend you do too. It is good habit to get into, to make it clear that this is an expression that generates a value.</p>
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<p>The <code>== 'Gold'</code> <em>comparison</em> is asking a question. It is asking that question of a vector, and the vector contains multiple values. R treats this comparison as asking the question <em>of each element in the vector</em>. We get an answer for the question <em>for each element</em>. The answer for position <span class="r">1</span> is <code>TRUE</code> if the element at position <span class="r">1</span> is equal to <code>'Gold'</code> and <code>FALSE</code> otherwise, and so on, for positions <span class="r">2</span>, <span class="r">3</span> and so on. We started with 10 strings. After the comparison <code>== 'Gold'</code> we have 10 Boolean values, where a Boolean value can either be <code>TRUE</code> or <code>FALSE</code>.</p>
<!---
We need an illustration here of form:
True "Gold"
True "Gold"
False "No gold in chest 1, chest 2 never opened"
True "Gold"
False = "Silver" == 'Gold'
...
Any takers? Good techniques? Maybe generated Markdown table with transparent boerder for second and fourth column?
-->
<p>Now we have an array with <code>TRUE</code> for the “Gold” results and <code>FALSE</code> otherwise, we can count the number of “Gold” results by using <code>sum</code> on the vector. As you remember (<a href="resampling_with_code.html#sec-count-with-sum" class="quarto-xref"><span>Section 5.15</span></a>) <code>sum</code> counts <code>TRUE</code> as 1 and <code>FALSE</code> as 0, so the sum of the Boolean vector is just the number of <code>TRUE</code> values in the vector — the count that we need.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The number of True values — so the number of "Gold" chests.</span></span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(were_gold)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 3</code></pre>
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</section>
<section id="sec-ships-booleans" class="level2" data-number="10.5">
<h2 data-number="10.5" class="anchored" data-anchor-id="sec-ships-booleans"><span class="header-section-number">10.5</span> Boolean vectors and another take on the ships problem</h2>
<p>If we are doing a physical simulation, we usually want to finish up all the work for the trial during the trial, so we have one outcome from the trial. This makes it easier to tally up the results in the end.</p>
<p>We have no such constraint when we are using code, so it is sometimes easier to record several results from the trial, and do the final combinations and tallies at the end. We will show you what we mean with a slight variation on the two-ships code you saw above.</p>
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Note 10.3: Notebook: Another approach to ships with gold and silver
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<p><a class="notebook-link" href="notebooks/gold_silver_booleans.Rmd">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=gold_silver_booleans.ipynb">Interact</a></p>
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<div class="nb-start" name="gold_silver_booleans" title="Another approach to ships with gold and silver">
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<p>Notice that the first part of the code is identical to the first approach to this problem. There are two key differences — see the comments for an explanation.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The 3 buckets, each representing two chests on a ship.</span></span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a><span class="co"># As before.</span></span>
<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a>bucket1 <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">'Gold'</span>, <span class="st">'Gold'</span>) <span class="co"># Chests in first ship.</span></span>
<span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a>bucket2 <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">'Gold'</span>, <span class="st">'Silver'</span>) <span class="co"># Chests in second ship.</span></span>
<span id="cb21-5"><a href="#cb21-5" aria-hidden="true" tabindex="-1"></a>bucket3 <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">'Silver'</span>, <span class="st">'Silver'</span>) <span class="co"># Chests in third ship.</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Here is where the difference starts. We are now going to fill in</span></span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a><span class="co"># the result for the first chest _and_ the result for the second chest.</span></span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a><span class="co">#</span></span>
<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Later we will fill in all these values, so the string we put here</span></span>
<span id="cb22-5"><a href="#cb22-5" aria-hidden="true" tabindex="-1"></a><span class="co"># does not matter.</span></span>
<span id="cb22-6"><a href="#cb22-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-7"><a href="#cb22-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Whether the first chest was Gold or Silver.</span></span>
<span id="cb22-8"><a href="#cb22-8" aria-hidden="true" tabindex="-1"></a>first_chests <span class="ot"><-</span> <span class="fu">rep</span>(<span class="st">'To be announced'</span>, <span class="dv">10000</span>)</span>
<span id="cb22-9"><a href="#cb22-9" aria-hidden="true" tabindex="-1"></a>second_chests <span class="ot"><-</span> <span class="fu">rep</span>(<span class="st">'To be announced'</span>, <span class="dv">10000</span>)</span>
<span id="cb22-10"><a href="#cb22-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-11"><a href="#cb22-11" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">10000</span>) {</span>
<span id="cb22-12"><a href="#cb22-12" aria-hidden="true" tabindex="-1"></a> <span class="co"># Select a ship at random from the three ships.</span></span>
<span id="cb22-13"><a href="#cb22-13" aria-hidden="true" tabindex="-1"></a> <span class="co"># As before.</span></span>
<span id="cb22-14"><a href="#cb22-14" aria-hidden="true" tabindex="-1"></a> ship_no <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>, <span class="at">size=</span><span class="dv">1</span>)</span>
<span id="cb22-15"><a href="#cb22-15" aria-hidden="true" tabindex="-1"></a> <span class="co"># Get the chests from this ship.</span></span>
<span id="cb22-16"><a href="#cb22-16" aria-hidden="true" tabindex="-1"></a> <span class="co"># As before.</span></span>
<span id="cb22-17"><a href="#cb22-17" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (ship_no <span class="sc">==</span> <span class="dv">1</span>) {</span>
<span id="cb22-18"><a href="#cb22-18" aria-hidden="true" tabindex="-1"></a> bucket <span class="ot"><-</span> bucket1</span>
<span id="cb22-19"><a href="#cb22-19" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb22-20"><a href="#cb22-20" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (ship_no <span class="sc">==</span> <span class="dv">2</span>) {</span>
<span id="cb22-21"><a href="#cb22-21" aria-hidden="true" tabindex="-1"></a> bucket <span class="ot"><-</span> bucket2</span>
<span id="cb22-22"><a href="#cb22-22" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb22-23"><a href="#cb22-23" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (ship_no <span class="sc">==</span> <span class="dv">3</span>) {</span>
<span id="cb22-24"><a href="#cb22-24" aria-hidden="true" tabindex="-1"></a> bucket <span class="ot"><-</span> bucket3</span>
<span id="cb22-25"><a href="#cb22-25" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb22-26"><a href="#cb22-26" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-27"><a href="#cb22-27" aria-hidden="true" tabindex="-1"></a> <span class="co"># As before.</span></span>
<span id="cb22-28"><a href="#cb22-28" aria-hidden="true" tabindex="-1"></a> shuffled <span class="ot"><-</span> <span class="fu">sample</span>(bucket)</span>
<span id="cb22-29"><a href="#cb22-29" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-30"><a href="#cb22-30" aria-hidden="true" tabindex="-1"></a> <span class="co"># Here is the big difference - we store the result for the first and second</span></span>
<span id="cb22-31"><a href="#cb22-31" aria-hidden="true" tabindex="-1"></a> <span class="co"># chests.</span></span>
<span id="cb22-32"><a href="#cb22-32" aria-hidden="true" tabindex="-1"></a> first_chests[i] <span class="ot"><-</span> shuffled[<span class="dv">1</span>]</span>
<span id="cb22-33"><a href="#cb22-33" aria-hidden="true" tabindex="-1"></a> second_chests[i] <span class="ot"><-</span> shuffled[<span class="dv">2</span>]</span>
<span id="cb22-34"><a href="#cb22-34" aria-hidden="true" tabindex="-1"></a>} <span class="co"># End loop, go back to beginning.</span></span>
<span id="cb22-35"><a href="#cb22-35" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-36"><a href="#cb22-36" aria-hidden="true" tabindex="-1"></a><span class="co"># We will do the calculation we need in the next cell. For now</span></span>
<span id="cb22-37"><a href="#cb22-37" aria-hidden="true" tabindex="-1"></a><span class="co"># just display the first 10 values.</span></span>
<span id="cb22-38"><a href="#cb22-38" aria-hidden="true" tabindex="-1"></a>ten_first_chests <span class="ot"><-</span> first_chests[<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>]</span>
<span id="cb22-39"><a href="#cb22-39" aria-hidden="true" tabindex="-1"></a><span class="fu">message</span>(<span class="st">'The first 10 values of "first_chests:'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>The first 10 values of "first_chests:</code></pre>
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<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(ten_first_chests)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code> [1] "Gold" "Silver" "Silver" "Gold" "Gold" "Gold" "Gold" "Gold"
[9] "Gold" "Gold" </code></pre>
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<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a>ten_second_chests <span class="ot"><-</span> second_chests[<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>]</span>
<span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a><span class="fu">message</span>(<span class="st">'The first 10 values of "second_chests:'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>The first 10 values of "second_chests:</code></pre>
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<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(ten_second_chests)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code> [1] "Gold" "Silver" "Silver" "Gold" "Silver" "Gold" "Silver" "Gold"
[9] "Silver" "Gold" </code></pre>
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<p>In this variant, we recorded the type of first chest for each trial (“Gold” or “Silver”), and the type of second chest of the second chest (“Gold” or “Silver”).</p>
<p><strong>We would like to count the number of times there was “Gold” in the first chest <em>and</em> “Gold” in the second.</strong></p>
</section>
<section id="sec-combine-booleans" class="level2" data-number="10.6">
<h2 data-number="10.6" class="anchored" data-anchor-id="sec-combine-booleans">10.6 Combining Boolean arrays</h2>
<p>We can do the count we need by <em>combining</em> the Boolean vectors with the <code>&</code> operator. <code>&</code> combines Boolean vectors with a <em>logical and</em>. <em>Logical and</em> is a rule for combining two Boolean values, where the rule is: the result is <code>TRUE</code> if the first value is <code>TRUE</code> <em>and</em> the second value if <code>TRUE</code>.</p>
<p>Here we use the <code>&</code> <em>operator</em> to combine some Boolean values on the left and right of the operator:</p>
<p>Above you saw that the <code>==</code> operator (as in <code>== 'Gold'</code>), when applied to vectors, asks the question of every element in the vector.</p>
<p>First make the Boolean vectors.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a>ten_first_gold <span class="ot"><-</span> ten_first_chests <span class="sc">==</span> <span class="st">'Gold'</span></span>
<span id="cb30-2"><a href="#cb30-2" aria-hidden="true" tabindex="-1"></a><span class="fu">message</span>(<span class="st">"Ten first == 'Gold'"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>Ten first == 'Gold'</code></pre>
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<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(ten_first_gold)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code> [1] TRUE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE</code></pre>
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<div class="sourceCode cell-code" id="cb34"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a>ten_second_gold <span class="ot"><-</span> ten_second_chests <span class="sc">==</span> <span class="st">'Gold'</span></span>
<span id="cb34-2"><a href="#cb34-2" aria-hidden="true" tabindex="-1"></a><span class="fu">message</span>(<span class="st">"Ten second == 'Gold'"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>Ten second == 'Gold'</code></pre>
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<div class="sourceCode cell-code" id="cb36"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(ten_second_gold)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code> [1] TRUE FALSE FALSE TRUE FALSE TRUE FALSE TRUE FALSE TRUE</code></pre>
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<p>Now let us use <code>&</code> to combine Boolean vectors:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb38"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a>ten_both <span class="ot"><-</span> (ten_first_gold <span class="sc">&</span> ten_second_gold)</span>
<span id="cb38-2"><a href="#cb38-2" aria-hidden="true" tabindex="-1"></a>ten_both</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code> [1] TRUE FALSE FALSE TRUE FALSE TRUE FALSE TRUE FALSE TRUE</code></pre>
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<p>Notice that R does the comparison <em>elementwise</em> — element by element.</p>
<p>You saw that when we did <code>second_chests == 'Gold'</code> this had the effect of asking the <code>== 'Gold'</code> question of <em>each element</em>, so there will be one answer per element in <code>second_chests</code>. In that case there was a vector to the <em>left</em> of <code>==</code> and a single value to the <em>right</em>. We were comparing a vector to a value.</p>
<p>Here we are asking the <code>&</code> question of <code>ten_first_gold</code> and <code>ten_second_gold</code>. Here there is a vector to the <em>left</em> and a vector to the <em>right</em>. We are asking the <code>&</code> question 10 times, but the first question we are asking is:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb40"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a><span class="co"># First question, giving first element of result.</span></span>
<span id="cb40-2"><a href="#cb40-2" aria-hidden="true" tabindex="-1"></a>(ten_first_gold[<span class="dv">1</span>] <span class="sc">&</span> ten_second_gold[<span class="dv">1</span>])</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] TRUE</code></pre>
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<p>The second question is:</p>
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<div class="sourceCode cell-code" id="cb42"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Second question, giving second element of result.</span></span>
<span id="cb42-2"><a href="#cb42-2" aria-hidden="true" tabindex="-1"></a>(ten_first_gold[<span class="dv">2</span>] <span class="sc">&</span> ten_second_gold[<span class="dv">2</span>])</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] FALSE</code></pre>
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<p>and so on. We have ten elements on <em>each side</em>, and 10 answers, giving a vector (<code>ten_both</code>) of 10 elements. Each element in <code>ten_both</code> is the answer to the <code>&</code> question for the elements at the corresponding positions in <code>ten_first_gold</code> and <code>ten_second_gold</code>.</p>
<p>We could also create the Boolean vectors and do the <code>&</code> operation all in one step, like this:</p>
<!---
Another illustration here, of form (matching the text):
True True True
False False True
False False True
True True False
False = False & True
...
-->
<p>Remember, we wanted the answer to the question: how many trials had “Gold” in the first chest <em>and</em> “Gold” in the second. We can answer that question for the first 10 trials with <code>sum</code>:</p>
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<div class="sourceCode cell-code" id="cb44"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a>n_ten_both <span class="ot"><-</span> <span class="fu">sum</span>(ten_both)</span>
<span id="cb44-2"><a href="#cb44-2" aria-hidden="true" tabindex="-1"></a>n_ten_both</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 5</code></pre>
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<p>We can answer the same question for <em>all</em> the trials, in the same way:</p>
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<div class="sourceCode cell-code" id="cb46"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a>first_gold <span class="ot"><-</span> first_chests <span class="sc">==</span> <span class="st">'Gold'</span></span>
<span id="cb46-2"><a href="#cb46-2" aria-hidden="true" tabindex="-1"></a>second_gold <span class="ot"><-</span> second_chests <span class="sc">==</span> <span class="st">'Gold'</span></span>
<span id="cb46-3"><a href="#cb46-3" aria-hidden="true" tabindex="-1"></a>n_both_gold <span class="ot"><-</span> <span class="fu">sum</span>(first_gold <span class="sc">&</span> second_gold)</span>
<span id="cb46-4"><a href="#cb46-4" aria-hidden="true" tabindex="-1"></a>n_both_gold</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 3328</code></pre>
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<p>We could also do the same calculation all in one line:</p>
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<div class="sourceCode cell-code" id="cb48"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb48-1"><a href="#cb48-1" aria-hidden="true" tabindex="-1"></a>n_both_gold <span class="ot"><-</span> <span class="fu">sum</span>((first_chests <span class="sc">==</span> <span class="st">'Gold'</span>) <span class="sc">&</span> (second_chests <span class="sc">==</span> <span class="st">'Gold'</span>))</span>
<span id="cb48-2"><a href="#cb48-2" aria-hidden="true" tabindex="-1"></a>n_both_gold</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 3328</code></pre>
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<p>We can then count all the ships where the first chest was gold:</p>
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<div class="sourceCode cell-code" id="cb50"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a>n_first_gold <span class="ot"><-</span> <span class="fu">sum</span>(first_chests <span class="sc">==</span> <span class="st">'Gold'</span>)</span>
<span id="cb50-2"><a href="#cb50-2" aria-hidden="true" tabindex="-1"></a>n_first_gold</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 5021</code></pre>
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<p>The final calculation is the proportion of second chests that are gold, given the first chest was also gold:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb52"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb52-1"><a href="#cb52-1" aria-hidden="true" tabindex="-1"></a>p_g_given_g <span class="ot"><-</span> n_both_gold <span class="sc">/</span> n_first_gold</span>
<span id="cb52-2"><a href="#cb52-2" aria-hidden="true" tabindex="-1"></a>p_g_given_g</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 0.663</code></pre>
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<p>Of course we won’t get exactly the same results from the two simulations, in the same way that we won’t get exactly the same results from any two runs of the same simulation, because of the random values we are using. But the logic for the two simulations are the same, and we are doing many trials (10,000), so the results will be very similar.</p>
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