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<h2 id="toc-title">Table of contents</h2>
<ul>
<li><a href="#introduction" id="toc-introduction" class="nav-link active" data-scroll-target="#introduction"><span class="header-section-number">13.1</span> Introduction</a></li>
<li><a href="#some-building-block-programs" id="toc-some-building-block-programs" class="nav-link" data-scroll-target="#some-building-block-programs"><span class="header-section-number">13.2</span> Some building-block programs</a></li>
<li><a href="#problems-in-finite-universes" id="toc-problems-in-finite-universes" class="nav-link" data-scroll-target="#problems-in-finite-universes"><span class="header-section-number">13.3</span> Problems in finite universes</a>
<ul class="collapse">
<li><a href="#sec-four-girls-one-boy" id="toc-sec-four-girls-one-boy" class="nav-link" data-scroll-target="#sec-four-girls-one-boy"><span class="header-section-number">13.3.1</span> Example: four girls and one boy</a></li>
<li><a href="#sec-five-spades-four-clubs" id="toc-sec-five-spades-four-clubs" class="nav-link" data-scroll-target="#sec-five-spades-four-clubs"><span class="header-section-number">13.3.2</span> Example: Five spades and four clubs in a bridge hand</a></li>
<li><a href="#sec-fifteen-bridge" id="toc-sec-fifteen-bridge" class="nav-link" data-scroll-target="#sec-fifteen-bridge"><span class="header-section-number">13.3.3</span> Example: a total of fifteen points in a bridge hand</a></li>
<li><a href="#example-four-girls-then-one-boy-from-25-girls-and-25-boys" id="toc-example-four-girls-then-one-boy-from-25-girls-and-25-boys" class="nav-link" data-scroll-target="#example-four-girls-then-one-boy-from-25-girls-and-25-boys"><span class="header-section-number">13.3.4</span> Example: Four girls then one boy from 25 girls and 25 boys</a></li>
<li><a href="#example-repeat-pairings-from-random-pairing" id="toc-example-repeat-pairings-from-random-pairing" class="nav-link" data-scroll-target="#example-repeat-pairings-from-random-pairing"><span class="header-section-number">13.3.5</span> Example: repeat pairings from random pairing</a></li>
<li><a href="#example-matching-santa-hats" id="toc-example-matching-santa-hats" class="nav-link" data-scroll-target="#example-matching-santa-hats"><span class="header-section-number">13.3.6</span> Example: Matching Santa Hats</a></li>
<li><a href="#example-twenty-executives-assigned-to-two-divisions-of-a-firm" id="toc-example-twenty-executives-assigned-to-two-divisions-of-a-firm" class="nav-link" data-scroll-target="#example-twenty-executives-assigned-to-two-divisions-of-a-firm"><span class="header-section-number">13.3.7</span> Example: Twenty executives assigned to two divisions of a firm</a></li>
<li><a href="#example-executives-moving" id="toc-example-executives-moving" class="nav-link" data-scroll-target="#example-executives-moving"><span class="header-section-number">13.3.8</span> Example: Executives Moving</a></li>
<li><a href="#example-state-liquor-systems-again" id="toc-example-state-liquor-systems-again" class="nav-link" data-scroll-target="#example-state-liquor-systems-again"><span class="header-section-number">13.3.9</span> Example: State Liquor Systems Again</a></li>
<li><a href="#sec-five-spades-four-girls" id="toc-sec-five-spades-four-girls" class="nav-link" data-scroll-target="#sec-five-spades-four-girls"><span class="header-section-number">13.3.10</span> Example: Five or More Spades in One Bridge Hand; Four Girls and a Boy</a></li>
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<li><a href="#summary" id="toc-summary" class="nav-link" data-scroll-target="#summary"><span class="header-section-number">13.4</span> Summary</a></li>
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<h1 class="title"><span id="sec-finite-universes" class="quarto-section-identifier"><span class="chapter-number">13</span> <span class="chapter-title">Probability Theory, Part 4: Estimating Probabilities from Finite Universes</span></span></h1>
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</header>
<section id="introduction" class="level2" data-number="13.1">
<h2 data-number="13.1" class="anchored" data-anchor-id="introduction"><span class="header-section-number">13.1</span> Introduction</h2>
<p>The examples in <a href="probability_theory_3.html" class="quarto-xref"><span>Chapter 12</span></a> dealt with <em>infinite universes</em>, in which the probability of a given simple event is unaffected by the outcome of the previous simple event. But now we move on to finite universes, situations in which you begin with a <em>given set of objects</em> whose number is not enormous — say, a total of two, or two hundred, or two thousand. If we liken such a situation to a bucket containing balls of different colors each with a number on it, we are interested in the probability of drawing various sets of numbered and colored balls from the bucket on the condition that we <em>do not replace</em> balls after they are drawn.</p>
<p>In the cases addressed in this chapter, it is important to remember that the single events no longer are independent of each other. A typical situation in which sampling without replacement occurs is when items are chosen from a finite universe — for example, when children are selected randomly from a classroom. If the class has five boys and five girls, and if you were to choose three girls in a row, then the chance of selecting a fourth girl on the next choice obviously is lower than the chance that you would pick a girl on the first selection.</p>
<p>The key to dealing with this type of problem is the same as with earlier problems: You must choose a simulation procedure that produces simple events having the same probabilities as the simple events in the actual problem involving sampling without replacement. That is, you must make sure that your simulation does not allow duplication of events that have already occurred. The easiest way to sample without replacement with resampling techniques is by simply ignoring an outcome if it has already occurred.</p>
<p>Examples <a href="#sec-four-girls-one-boy" class="quarto-xref"><span>Section 13.3.1</span></a> through <a href="#sec-five-spades-four-girls" class="quarto-xref"><span>Section 13.3.10</span></a> deal with some of the more important sorts of questions one may ask about drawings without replacement from such an urn. To get an overview, I suggest that you read over the summaries (in <strong>bold</strong>) introducing examples <a href="#sec-four-girls-one-boy" class="quarto-xref"><span>Section 13.3.1</span></a> to <a href="#sec-five-spades-four-girls" class="quarto-xref"><span>Section 13.3.10</span></a> before beginning to work through the examples themselves.</p>
<p>This chapter also revisits the general procedure used in solving problems in probability and statistics with simulation, here in connection with problems involving a finite universe. The steps that one follows in simulating the behavior of a universe of interest are set down in such fashion that one may, by random drawings, deduce the probability of various events. Having had by now the experience of working through the problems in <a href="probability_theory_1b.html" class="quarto-xref"><span>Chapter 9</span></a> and <a href="probability_theory_3.html" class="quarto-xref"><span>Chapter 12</span></a>, the reader should have a solid basis to follow the description of the general procedure which then helps in dealing with specific problems.</p>
<p>Let us begin by describing some of the major sorts of problems with the aid of a bucket with six balls.</p>
</section>
<section id="some-building-block-programs" class="level2" data-number="13.2">
<h2 data-number="13.2" class="anchored" data-anchor-id="some-building-block-programs"><span class="header-section-number">13.2</span> Some building-block programs</h2>
<p><strong>Case 1.</strong> Each of six balls is labeled with a number between “1” and “6.” We ask: What is the probability of choosing balls 1, 2, and 3 <em>in that order</em> if we choose three balls without replacement? <a href="#fig-success_case1" class="quarto-xref">Figure <span>13.1</span></a> diagrams the events we consider “success.”</p>
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<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-success_case1-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 13.1: The Event Classified as “Success” for Case 1
</figcaption>
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<p><strong>Case 2.</strong> We begin with the same bucket as in Case 1, but now ask the probability of choosing balls 1, 2, and 3 <em>in any order</em> if we choose three balls without replacement. <a href="#fig-success_case2" class="quarto-xref">Figure <span>13.2</span></a> diagrams two of the events we consider success. These possibilities include that which is shown in <a href="#fig-success_case1" class="quarto-xref">Figure <span>13.1</span></a> above, plus other possibilities.</p>
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Figure 13.2: An Incomplete List of the Events Classified as “Success” for Case 2
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<p><strong>Case 3.</strong> The odd-numbered balls “1,” “3,” and “5,” are painted red and the even-numbered balls “2,” “4,” and “6” are painted black. What is the probability of getting a red ball and then a black ball in that order? Some possibilities are illustrated in <a href="#fig-success_case3" class="quarto-xref">Figure <span>13.3</span></a>, which includes the possibility shown in <a href="#fig-success_case1" class="quarto-xref">Figure <span>13.1</span></a>. It also includes <em>some but not</em> all possibilities found in <a href="#fig-success_case2" class="quarto-xref">Figure <span>13.2</span></a>; for example, <a href="#fig-success_case2" class="quarto-xref">Figure <span>13.2</span></a> includes choosing balls 2, 3 and 1 in that order, but <a href="#fig-success_case3" class="quarto-xref">Figure <span>13.3</span></a> does not.</p>
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<img src="diagrams/success_case3.svg" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%">
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<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-success_case3-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 13.3: An Incomplete List of the Events Classified as “Success” for Case 3
</figcaption>
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<p><strong>Case 4.</strong> What is the probability of getting two red balls and one black ball in any order?</p>
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<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-success_case4-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 13.4: An Incomplete List of the Events Classified as “Success” for Case 4
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<p><strong>Case 5.</strong> Various questions about <em>matching</em> may be asked with respect to the six balls. For example, what is the probability of getting ball 1 on the first draw <em>or</em> ball 2 on the second draw <em>or</em> ball 3 on the third draw? (<a href="#fig-success_case5" class="quarto-xref">Figure <span>13.5</span></a>) Or, what is the probability of getting all balls on the draws corresponding to their numbers?</p>
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<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-success_case5-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 13.5: An Incomplete List of the Events Classified as “Success” for Case 5
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</section>
<section id="problems-in-finite-universes" class="level2" data-number="13.3">
<h2 data-number="13.3" class="anchored" data-anchor-id="problems-in-finite-universes"><span class="header-section-number">13.3</span> Problems in finite universes</h2>
<section id="sec-four-girls-one-boy" class="level3" data-number="13.3.1">
<h3 data-number="13.3.1" class="anchored" data-anchor-id="sec-four-girls-one-boy"><span class="header-section-number">13.3.1</span> Example: four girls and one boy</h3>
<p><strong>What is the probability of selecting four girls and one boy when selecting five students from any group of twenty-five girls and twenty-five boys?</strong> This is an example of sampling without replacement when there are two outcomes and the order does not matter.</p>
<p>The important difference between this example and the infinite-universe examples in the prior chapter is that the probability of obtaining a boy or a girl in a single simple event <em>differs</em> from one event to the next in this example, whereas it stays the same when the sampling is with replacement. To illustrate, the probability of a girl is .5 (25 out of 50) when the first student is chosen, but the probability of a girl is either 25/49 or 24/49 when the second student is chosen, depending on whether a boy or a girl was chosen on the first pick. Or after, say, three girls and one boy are picked, the probability of getting a girl on the next choice is (28-3)/(50-4) = 22/46 which is clearly not equal to .5.</p>
<p>As always, we must create a satisfactory analog to the process whose probability we want to learn. In this case, we can use a deck of 50 cards, half red and half black, and deal out five cards <em>without replacing them</em> after each card is dealt; this simulates the choice of five students from among the fifty.</p>
<p>We can no longer use our procedure from before. If we designated “1-25” as being girls and “26-50” as being boys and then proceeded to draw random numbers, the probability of a girl would be the same on each pick.</p>
<p>At this point, it is important to note that — for this particular problem — we do not need to distinguish between particular girls (or boys). That is, it does not matter <em>which</em> girl (or boy) is selected in a given trial. Nor did we pay attention to the <em>order</em> in which we selected girls or boys. This is an instance of Case 4 discussed above. Subsequent problems will deal with situations where the order of selection, and the particular individuals, do matter.</p>
<p>Our approach then is to mimic having the class in front of us: an array of 50 strings, half of the entries ‘boy’ and the other half ‘girl’. We then shuffle the class (the array), and choose the first N students (strings).</p>
<ul>
<li><strong>Step 1.</strong> Create a list with 50 labels, half ‘boy’ and half ‘girl’.</li>
<li><strong>Step 2.</strong> Shuffle the class and select five students. Count whether there are four labels equal ‘girl’. If so, write “yes,” otherwise “no”.</li>
<li><strong>Step 3.</strong> Repeat step 2, say, 10,000 times, and count the proportion “yes”, which estimates the probability sought.</li>
</ul>
<p>The results of a few experimental trials are shown in <a href="#tbl-four-girls-one-boy" class="quarto-xref">Table <span>13.1</span></a>.</p>
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<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-four-girls-one-boy-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 13.1: A few experimental trials of four girls and one boy
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<td><strong>Experiment</strong></td>
<td><strong>Strings Chosen</strong></td>
<td><strong>Success?</strong></td>
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<td><pre><code> 1</code></pre></td>
<td>‘girl’, ‘boy’, ‘boy’, ‘girl’, ‘boy’</td>
<td>No</td>
</tr>
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<td><pre><code> 2</code></pre></td>
<td>‘boy’, ‘girl’, ‘girl’, ‘girl’, ‘girl’</td>
<td>Yes</td>
</tr>
<tr class="even">
<td><pre><code> 3</code></pre></td>
<td>‘girl, ’girl’, ‘girl’, ‘boy’, ‘girl’</td>
<td>Yes</td>
</tr>
</tbody>
</table>
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<p>A solution to this problem with Python is presented below.</p>
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Note 13.1: Notebook: Four girls and one boy
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<p><a class="notebook-link" href="notebooks/four_girls_one_boy.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=four_girls_one_boy.ipynb">Interact</a></p>
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<div class="nb-start" name="four_girls_one_boy" title="Four girls and one boy">
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>N <span class="op">=</span> <span class="dv">10000</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>trial_results <span class="op">=</span> np.zeros(N)</span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Constitute the set of 25 girls and 25 boys.</span></span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a>whole_class <span class="op">=</span> np.repeat([<span class="st">'girl'</span>, <span class="st">'boy'</span>], [<span class="dv">25</span>, <span class="dv">25</span>])</span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat the following steps N times.</span></span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N):</span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a> <span class="co"># Shuffle the numbers</span></span>
<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a> shuffled <span class="op">=</span> rnd.permuted(whole_class)</span>
<span id="cb5-12"><a href="#cb5-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-13"><a href="#cb5-13" aria-hidden="true" tabindex="-1"></a> <span class="co"># Take the first 5 numbers, call them c.</span></span>
<span id="cb5-14"><a href="#cb5-14" aria-hidden="true" tabindex="-1"></a> c <span class="op">=</span> shuffled[:<span class="dv">5</span>]</span>
<span id="cb5-15"><a href="#cb5-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-16"><a href="#cb5-16" aria-hidden="true" tabindex="-1"></a> <span class="co"># Count how many girls there are, put the result in d.</span></span>
<span id="cb5-17"><a href="#cb5-17" aria-hidden="true" tabindex="-1"></a> d <span class="op">=</span> np.<span class="bu">sum</span>(c <span class="op">==</span> <span class="st">'girl'</span>)</span>
<span id="cb5-18"><a href="#cb5-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-19"><a href="#cb5-19" aria-hidden="true" tabindex="-1"></a> <span class="co"># Keep track of each trial result in z.</span></span>
<span id="cb5-20"><a href="#cb5-20" aria-hidden="true" tabindex="-1"></a> trial_results[i] <span class="op">=</span> d</span>
<span id="cb5-21"><a href="#cb5-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-22"><a href="#cb5-22" aria-hidden="true" tabindex="-1"></a> <span class="co"># End the experiment, go back and repeat until all 1000 trials are</span></span>
<span id="cb5-23"><a href="#cb5-23" aria-hidden="true" tabindex="-1"></a> <span class="co"># complete.</span></span>
<span id="cb5-24"><a href="#cb5-24" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-25"><a href="#cb5-25" aria-hidden="true" tabindex="-1"></a><span class="co"># Count the number of times we got four girls, put the result in k.</span></span>
<span id="cb5-26"><a href="#cb5-26" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> np.<span class="bu">sum</span>(trial_results <span class="op">==</span> <span class="dv">4</span>)</span>
<span id="cb5-27"><a href="#cb5-27" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-28"><a href="#cb5-28" aria-hidden="true" tabindex="-1"></a><span class="co"># Convert to a proportion.</span></span>
<span id="cb5-29"><a href="#cb5-29" aria-hidden="true" tabindex="-1"></a>kk <span class="op">=</span> k <span class="op">/</span> N</span>
<span id="cb5-30"><a href="#cb5-30" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-31"><a href="#cb5-31" aria-hidden="true" tabindex="-1"></a><span class="co"># Print the result.</span></span>
<span id="cb5-32"><a href="#cb5-32" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(kk)</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>0.1505</code></pre>
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</div>
<p>We can also find the probabilities of other outcomes from a histogram of trial results obtained with the following command:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Import the plotting package.</span></span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Do histogram, with one bin for each possible number.</span></span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a>plt.hist(trial_results, bins<span class="op">=</span><span class="bu">range</span>(<span class="dv">7</span>), align<span class="op">=</span><span class="st">'left'</span>, rwidth<span class="op">=</span><span class="fl">0.75</span>)</span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">'# of girls'</span>)<span class="op">;</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
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<p><img src="probability_theory_4_finite_files/figure-html/unnamed-chunk-4-1.png" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%"></p>
</figure>
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<p>In the resulting histogram we can see that in 15 percent of the trials, 4 of the 5 selected were girls.</p>
<p>It should be noted that for this problem — as for most other problems — there are several other resampling procedures that will also do the job correctly.</p>
<p>In analytic probability theory this problem is worked with a formula for “combinations.”</p>
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End of notebook: Four girls and one boy
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<div class="callout-body-container callout-body">
<p><code>four_girls_one_boy</code> starts at <a href="#nte-four_girls_one_boy" class="quarto-xref">Note <span>13.1</span></a>.</p>
</div>
</div>
</section>
<section id="sec-five-spades-four-clubs" class="level3" data-number="13.3.2">
<h3 data-number="13.3.2" class="anchored" data-anchor-id="sec-five-spades-four-clubs"><span class="header-section-number">13.3.2</span> Example: Five spades and four clubs in a bridge hand</h3>
<div id="nte-five_spades_four_clubs" class="callout callout-style-default callout-note callout-titled">
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Note 13.2: Notebook: Five spades and four clubs
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<div class="callout-body-container callout-body">
<div class="nb-links">
<p><a class="notebook-link" href="notebooks/five_spades_four_clubs.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=five_spades_four_clubs.ipynb">Interact</a></p>
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<div class="nb-start" name="five_spades_four_clubs" title="Five spades and four clubs">
</div>
<p><strong>This is an example of multiple-outcome sampling without replacement, order does not matter</strong>.</p>
<p>The problem is similar to the example in <a href="#sec-four-girls-one-boy" class="quarto-xref"><span>Section 13.3.1</span></a>, except that now there are four equally-likely outcomes instead of only two. A Python solution is:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Constitute the deck of 52 cards.</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat the suit names 13 times each, to make a 52 card deck.</span></span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>deck <span class="op">=</span> np.repeat([<span class="st">'spade'</span>, <span class="st">'club'</span>, <span class="st">'diamond'</span>, <span class="st">'heart'</span>],</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a> [<span class="dv">13</span>, <span class="dv">13</span>, <span class="dv">13</span>, <span class="dv">13</span>])</span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the deck</span></span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a>deck</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>array(['spade', 'spade', 'spade', 'spade', 'spade', 'spade', 'spade',
'spade', 'spade', 'spade', 'spade', 'spade', 'spade', 'club',
'club', 'club', 'club', 'club', 'club', 'club', 'club', 'club',
'club', 'club', 'club', 'club', 'diamond', 'diamond', 'diamond',
'diamond', 'diamond', 'diamond', 'diamond', 'diamond', 'diamond',
'diamond', 'diamond', 'diamond', 'diamond', 'heart', 'heart',
'heart', 'heart', 'heart', 'heart', 'heart', 'heart', 'heart',
'heart', 'heart', 'heart', 'heart'], dtype='<U7')</code></pre>
</div>
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>N <span class="op">=</span> <span class="dv">10000</span></span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>trial_results <span class="op">=</span> np.zeros(N)</span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat the trial N times.</span></span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N):</span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a> <span class="co"># Shuffle the deck and draw 13 cards.</span></span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a> hand <span class="op">=</span> rnd.choice(deck, size<span class="op">=</span><span class="dv">13</span>, replace<span class="op">=</span><span class="va">False</span>)</span>
<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-10"><a href="#cb11-10" aria-hidden="true" tabindex="-1"></a> <span class="co"># Count the number of spades in "hand", put the result in "n_spades".</span></span>
<span id="cb11-11"><a href="#cb11-11" aria-hidden="true" tabindex="-1"></a> n_spades <span class="op">=</span> np.<span class="bu">sum</span>(hand <span class="op">==</span> <span class="st">'spade'</span>)</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="co"># If we have five spades, we'll continue on to count the clubs. If we don't</span></span>
<span id="cb11-14"><a href="#cb11-14" aria-hidden="true" tabindex="-1"></a> <span class="co"># have five spades, the number of clubs is irrelevant — we have not gotten</span></span>
<span id="cb11-15"><a href="#cb11-15" aria-hidden="true" tabindex="-1"></a> <span class="co"># the hand we are interested in.</span></span>
<span id="cb11-16"><a href="#cb11-16" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> n_spades <span class="op">==</span> <span class="dv">5</span>:</span>
<span id="cb11-17"><a href="#cb11-17" aria-hidden="true" tabindex="-1"></a> <span class="co"># Count the clubs, put the result in "n_clubs"</span></span>
<span id="cb11-18"><a href="#cb11-18" aria-hidden="true" tabindex="-1"></a> n_clubs <span class="op">=</span> np.<span class="bu">sum</span>(hand <span class="op">==</span> <span class="st">'club'</span>)</span>
<span id="cb11-19"><a href="#cb11-19" aria-hidden="true" tabindex="-1"></a> <span class="co"># Keep track of the number of clubs in each trial</span></span>
<span id="cb11-20"><a href="#cb11-20" aria-hidden="true" tabindex="-1"></a> trial_results[i] <span class="op">=</span> n_clubs</span>
<span id="cb11-21"><a href="#cb11-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-22"><a href="#cb11-22" aria-hidden="true" tabindex="-1"></a> <span class="co"># End one experiment, go back and repeat until all N trials are done.</span></span>
<span id="cb11-23"><a href="#cb11-23" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-24"><a href="#cb11-24" aria-hidden="true" tabindex="-1"></a><span class="co"># Count the number of trials where we got 4 clubs. This is the answer we want -</span></span>
<span id="cb11-25"><a href="#cb11-25" aria-hidden="true" tabindex="-1"></a><span class="co"># the number of hands out of 1000 with 5 spades and 4 clubs. (Recall that we</span></span>
<span id="cb11-26"><a href="#cb11-26" aria-hidden="true" tabindex="-1"></a><span class="co"># only counted the clubs if the hand already had 5 spades.)</span></span>
<span id="cb11-27"><a href="#cb11-27" aria-hidden="true" tabindex="-1"></a>n_5_and_4 <span class="op">=</span> np.<span class="bu">sum</span>(trial_results <span class="op">==</span> <span class="dv">4</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"># Convert to a proportion.</span></span>
<span id="cb11-30"><a href="#cb11-30" aria-hidden="true" tabindex="-1"></a>prop_5_and_4 <span class="op">=</span> n_5_and_4 <span class="op">/</span> N</span>
<span id="cb11-31"><a href="#cb11-31" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-32"><a href="#cb11-32" aria-hidden="true" tabindex="-1"></a><span class="co"># Print the result</span></span>
<span id="cb11-33"><a href="#cb11-33" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(prop_5_and_4)</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>0.0224</code></pre>
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<div class="nb-end">
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End of notebook: Five spades and four clubs
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<p><code>five_spades_four_clubs</code> starts at <a href="#nte-five_spades_four_clubs" class="quarto-xref">Note <span>13.2</span></a>.</p>
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</div>
</section>
<section id="sec-fifteen-bridge" class="level3" data-number="13.3.3">
<h3 data-number="13.3.3" class="anchored" data-anchor-id="sec-fifteen-bridge"><span class="header-section-number">13.3.3</span> Example: a total of fifteen points in a bridge hand</h3>
<div id="nte-fifteen_points_in_bridge" class="callout callout-style-default callout-note callout-titled">
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Note 13.3: Notebook: Fifteen points in a bridge hand
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<div class="nb-links">
<p><a class="notebook-link" href="notebooks/fifteen_points_in_bridge.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=fifteen_points_in_bridge.ipynb">Interact</a></p>
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<div class="nb-start" name="fifteen_points_in_bridge" title="Fifteen points in a bridge hand">
</div>
<p>Let us assume that ace counts as 4, king = 3, queen = 2, and jack = 1.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Constitute a deck with 4 jacks (point value 1), 4 queens (value 2), 4</span></span>
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a><span class="co"># kings (value 3), 4 aces (value 4), and 36 other cards with no point</span></span>
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a><span class="co"># value</span></span>
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a>whole_deck <span class="op">=</span> np.repeat([<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>, <span class="dv">4</span>, <span class="dv">0</span>], [<span class="dv">4</span>, <span class="dv">4</span>, <span class="dv">4</span>, <span class="dv">4</span>, <span class="dv">36</span>])</span>
<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a>whole_deck</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>array([1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0])</code></pre>
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</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a>N <span class="op">=</span> <span class="dv">10000</span></span>
<span id="cb16-2"><a href="#cb16-2" aria-hidden="true" tabindex="-1"></a>trial_results <span class="op">=</span> np.zeros(N)</span>
<span id="cb16-3"><a href="#cb16-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb16-4"><a href="#cb16-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Do N trials.</span></span>
<span id="cb16-5"><a href="#cb16-5" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(N):</span>
<span id="cb16-6"><a href="#cb16-6" aria-hidden="true" tabindex="-1"></a> <span class="co"># Shuffle the deck of cards and draw 13</span></span>
<span id="cb16-7"><a href="#cb16-7" aria-hidden="true" tabindex="-1"></a> hand <span class="op">=</span> rnd.choice(whole_deck, size<span class="op">=</span><span class="dv">13</span>, replace<span class="op">=</span><span class="va">False</span>)</span>
<span id="cb16-8"><a href="#cb16-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb16-9"><a href="#cb16-9" aria-hidden="true" tabindex="-1"></a> <span class="co"># Total the points.</span></span>
<span id="cb16-10"><a href="#cb16-10" aria-hidden="true" tabindex="-1"></a> points <span class="op">=</span> np.<span class="bu">sum</span>(hand)</span>
<span id="cb16-11"><a href="#cb16-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb16-12"><a href="#cb16-12" aria-hidden="true" tabindex="-1"></a> <span class="co"># Keep score of the result.</span></span>
<span id="cb16-13"><a href="#cb16-13" aria-hidden="true" tabindex="-1"></a> trial_results[i] <span class="op">=</span> points</span>
<span id="cb16-14"><a href="#cb16-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb16-15"><a href="#cb16-15" aria-hidden="true" tabindex="-1"></a> <span class="co"># End one experiment, go back and repeat until all N trials are done.</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Produce a histogram of trial results.</span></span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a>plt.hist(trial_results, bins<span class="op">=</span><span class="bu">range</span>(<span class="dv">25</span>), align<span class="op">=</span><span class="st">'left'</span>, rwidth<span class="op">=</span><span class="fl">0.75</span>)</span>
<span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">'Points in bridge hands'</span>)<span class="op">;</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>From this histogram, we see that in about 4 percent of our trials we obtained a total of exactly 15 points. We can also compute this directly:</p>
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<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="co"># How many times did we have a hand with fifteen points?</span></span>
<span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> np.<span class="bu">sum</span>(trial_results <span class="op">==</span> <span class="dv">15</span>)</span>
<span id="cb18-3"><a href="#cb18-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb18-4"><a href="#cb18-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Convert to a proportion.</span></span>
<span id="cb18-5"><a href="#cb18-5" aria-hidden="true" tabindex="-1"></a>kk <span class="op">=</span> k <span class="op">/</span> N</span>
<span id="cb18-6"><a href="#cb18-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb18-7"><a href="#cb18-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result.</span></span>
<span id="cb18-8"><a href="#cb18-8" aria-hidden="true" tabindex="-1"></a>kk</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>np.float64(0.0431)</code></pre>
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End of notebook: Fifteen points in a bridge hand
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<p><code>fifteen_points_in_bridge</code> starts at <a href="#nte-fifteen_points_in_bridge" class="quarto-xref">Note <span>13.3</span></a>.</p>
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<section id="example-four-girls-then-one-boy-from-25-girls-and-25-boys" class="level3" data-number="13.3.4">
<h3 data-number="13.3.4" class="anchored" data-anchor-id="example-four-girls-then-one-boy-from-25-girls-and-25-boys"><span class="header-section-number">13.3.4</span> Example: Four girls then one boy from 25 girls and 25 boys</h3>
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Note 13.4: Notebook: Four girls then one boy from 25/25
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<p><a class="notebook-link" href="notebooks/four_girls_then_one_boy_25.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=four_girls_then_one_boy_25.ipynb">Interact</a></p>
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<p><strong>In this problem, order matters; we are sampling without replacement, with two outcomes, several of each item.</strong></p>
<p>What is the probability of getting an ordered series of <em>four girls and then one boy</em>, from a universe of 25 girls and 25 boys? This illustrates Case 3 above. Clearly we can use the same sampling mechanism as in the example <a href="#sec-four-girls-one-boy" class="quarto-xref"><span>Section 13.3.1</span></a>, but now we record “yes” for a smaller number of composite events.</p>
<p>We record “no” even if a single one boy is chosen but he is chosen 1st, 2nd, 3rd, or 4th, whereas in <a href="#sec-four-girls-one-boy" class="quarto-xref"><span>Section 13.3.1</span></a>, such outcomes are recorded as “yes”-es.</p>
<ul>
<li><strong>Step 1.</strong> Generate a class (array) of length 50, consisting of 25 strings valued “boy” and 25 strings valued “girl”.</li>
<li><strong>Step 2.</strong> Shuffle the class array, and select the first five elements.</li>
<li><strong>Step 3.</strong> If the first five elements are exactly <code>'girl', 'girl', 'girl', 'girl', 'boy'</code>, write “yes,” otherwise “no.”</li>
<li><strong>Step 4.</strong> Repeat steps 2 and 3, say, 10,000 times, and count the proportion of “yes” results, which estimates the probability sought.</li>
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<p>Let us start the single trial procedure like so:</p>
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<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Constitute the set of 25 girls and 25 boys.</span></span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a>whole_class <span class="op">=</span> np.repeat([<span class="st">'girl'</span>, <span class="st">'boy'</span>], [<span class="dv">25</span>, <span class="dv">25</span>])</span>
<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Shuffle the class into a random order.</span></span>
<span id="cb21-5"><a href="#cb21-5" aria-hidden="true" tabindex="-1"></a>shuffled <span class="op">=</span> rnd.permuted(whole_class)</span>
<span id="cb21-6"><a href="#cb21-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Take the first 5 class members, call them c.</span></span>
<span id="cb21-7"><a href="#cb21-7" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> shuffled[:<span class="dv">5</span>]</span>
<span id="cb21-8"><a href="#cb21-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result.</span></span>
<span id="cb21-9"><a href="#cb21-9" aria-hidden="true" tabindex="-1"></a>c</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array(['boy', 'girl', 'boy', 'girl', 'girl'], dtype='<U4')</code></pre>
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<p>Our next step (step 3) is to check whether <code>c</code> is exactly equal to the result of interest. The result of interest is:</p>
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<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The result we are looking for - four girls and then a boy.</span></span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a>result_of_interest <span class="op">=</span> np.repeat([<span class="st">'girl'</span>, <span class="st">'boy'</span>], [<span class="dv">4</span>, <span class="dv">1</span>])</span>
<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a>result_of_interest</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array(['girl', 'girl', 'girl', 'girl', 'boy'], dtype='<U4')</code></pre>
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<p>We can then use an array <em>comparison</em> with <code>==</code> to do an element by element (<em>elementwise</em>) check, asking whether the corresponding elements are equal:</p>
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<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="co"># A Boolean array, with True where corresponding elements are equal, False</span></span>
<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a><span class="co"># otherwise.</span></span>
<span id="cb25-3"><a href="#cb25-3" aria-hidden="true" tabindex="-1"></a>are_equal <span class="op">=</span> c <span class="op">==</span> result_of_interest</span>
<span id="cb25-4"><a href="#cb25-4" aria-hidden="true" tabindex="-1"></a>are_equal</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>array([False, True, False, True, False])</code></pre>
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<p>We are nearly finished with step 3 — it only remains to check whether <em>all</em> of the elements were equal, by checking whether <em>all</em> of the values in <code>are_equal</code> are <code>True</code>.</p>
<p>We know that there are 5 elements, so we could check whether there are 5 <code>True</code> values with <code>np.sum</code>:</p>
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<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Are there exactly 5 True values in `are_equal`?</span></span>
<span id="cb27-2"><a href="#cb27-2" aria-hidden="true" tabindex="-1"></a>np.<span class="bu">sum</span>(are_equal) <span class="op">==</span> <span class="dv">5</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>np.False_</code></pre>
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<p>Another way to ask the same question is by using the <code>np.all</code> function on <code>are_equal</code>. This returns <code>True</code> if <em>all</em> the elements in <code>are_equal</code> are <code>True</code>, and <code>False</code> otherwise.</p>
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