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<li><a href="#the-resampling-approach-in-action" id="toc-the-resampling-approach-in-action" class="nav-link active" data-scroll-target="#the-resampling-approach-in-action"><span class="header-section-number">2.1</span> The resampling approach in action</a>
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<li><a href="#randomness-from-physical-methods" id="toc-randomness-from-physical-methods" class="nav-link" data-scroll-target="#randomness-from-physical-methods"><span class="header-section-number">2.1.1</span> Randomness from physical methods</a></li>
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<li><a href="#sec-randomness-computer" id="toc-sec-randomness-computer" class="nav-link" data-scroll-target="#sec-randomness-computer"><span class="header-section-number">2.2</span> Randomness from your computer</a></li>
<li><a href="#solving-the-problem-using" id="toc-solving-the-problem-using" class="nav-link" data-scroll-target="#solving-the-problem-using"><span class="header-section-number">2.3</span> Solving the problem using R</a></li>
<li><a href="#sec-resamp-differs" id="toc-sec-resamp-differs" class="nav-link" data-scroll-target="#sec-resamp-differs"><span class="header-section-number">2.4</span> How resampling differs from the conventional approach</a></li>
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<h1 class="title"><span id="sec-resampling-method" class="quarto-section-identifier"><span class="chapter-number">2</span> <span class="chapter-title">The resampling method</span></span></h1>
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<p>This chapter is a brief introduction to the resampling method of solving problems in probability and statistics. We’re going to dive right in and solve a problem hands-on.</p>
<p>You will see that the resampling method is easy to understand and apply: all it requires is to understand the physical problem. You then simulate a statistical model of the physical problem with techniques that are intuitively obvious, and estimate the probability sought with repeated random sampling.</p>
<p>After finding a solution, we will look at the more conventional formulaic approach, and how that compares. Here’s the spoiler: it requires you to understand complex formulas, and to choose the correct one from many.</p>
<p>After reading this chapter, you will understand why we are excited about the resampling method, and why it will allow you to approach even even hard problems without knowing sophisticated statistic techniques.</p>
<section id="the-resampling-approach-in-action" class="level2" data-number="2.1">
<h2 data-number="2.1" class="anchored" data-anchor-id="the-resampling-approach-in-action"><span class="header-section-number">2.1</span> The resampling approach in action</h2>
<p>Recall the problem from section <a href="intro.html#sec-what-problems" class="quarto-xref"><span>Section 1.2</span></a> in which the contractor owns 16 ambulances:</p>
<blockquote class="blockquote">
<p><em>You are the manager and part owner</em> of one of several contractors providing ambulances to a hospital. You own 16 ambulances. Based on past experience, the chance that any one ambulance will be unfit for service on any given day is about one in ten. You want to know the chance on a particular day — tomorrow — that three or more of them will be out of action.</p>
</blockquote>
<p>The resampling approach produces the estimate as follows.</p>
<section id="randomness-from-physical-methods" class="level3" data-number="2.1.1">
<h3 data-number="2.1.1" class="anchored" data-anchor-id="randomness-from-physical-methods"><span class="header-section-number">2.1.1</span> Randomness from physical methods</h3>
<p>We collect 10 coins, and mark one of them with a pen or pencil or tape as being the coin that represents “out-of-order;” the other nine coins stand for “in operation”. For any one ambulance, this set of 10 coins provides a “model” for the one-in-ten chance — a probability of .10 (10 percent) — of it being out of order on a given day. We put the coins into a little jar or bucket.</p>
<p>For ambulance #1, we draw a single coin from the bucket. This coin represents whether that ambulance is going to be broken tomorrow. After replacing the coin and shaking the bucket, we repeat the same procedure for ambulance #2, ambulance #3 and so forth. Having repeated the procedure <em>16</em> times, we now have a representation of all ambulances for a single day.</p>
<p>We can now repeat this whole process as many times as we like: each time, we come up with a representation for a different day, telling us how many ambulances will be out-of-service on that day.</p>
<p>After collecting evidence for, say, 50 experimental days we determine the proportion of the experimental days on which three or more ambulances are out of order. That proportion is an estimate of the probability that three or more ambulances will be out of order on a given day — the answer we seek. This procedure is an example of Monte Carlo simulation, which is the heart of the resampling method of statistical estimation.</p>
<p>A more direct way to answer this question would be to examine the firm’s actual records for the past 100 days or, better, 500 days (if that’s available) to determine how many days had three or more ambulances out of order. But the resampling procedure described above gives us an estimate even if we do not have such long-term information. This is realistic; it is frequently the case in the real world that we must make estimates on the basis of insufficient history about an event.</p>
<p>A quicker resampling method than the coins could be obtained with 16 ten-sided dice or spinners (like those found in Dungeons & Dragons games). For each die, we identify one of its ten sides as “out-of-order”.</p>
<p>Standard 10-sided dice have the numbers 0 through 9 on their faces, rather than 1 through 10. <a href="#fig-one-d10s" class="quarto-xref">Figure <span>2.1</span></a> shows a standard 10-sided die:</p>
<div id="fig-one-d10s" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-one-d10s-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-one-d10s-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 2.1: 10-sided die
</figcaption>
</figure>
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<p>We decide, arbitrarily, that the 9 side means “out-of-order”. We could even put a little bit of paint on the 9 side to remind us. The die represents an ambulance. If we roll the die, and get this face, this indicates that the ambulance was out of order. If we get any of the other faces — 0 through 8 — this ambulance was in working order. A single throw of all 16 dice will be our experimental trial that represents a single day; we just have to count whether three or more ambulances turn up “out of order”. <a href="#fig-16-d10s" class="quarto-xref">Figure <span>2.2</span></a> show the result of one trial — throwing 16 dice:</p>
<div id="fig-16-d10s" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-16-d10s-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="images/16_d10s.jpg" class="img-fluid figure-img">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-16-d10s-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 2.2: 16 10-sided dice
</figcaption>
</figure>
</div>
<p>As you can see, the trial in <a href="#fig-16-d10s" class="quarto-xref">Figure <span>2.2</span></a> gave us a single 9, so there was only one ambulance out of order.</p>
<p>In a hundred throws of the 16 dice — which probably takes less than 5 minutes — we can get a fast and reasonably accurate answer to our question.</p>
</section>
</section>
<section id="sec-randomness-computer" class="level2" data-number="2.2">
<h2 data-number="2.2" class="anchored" data-anchor-id="sec-randomness-computer"><span class="header-section-number">2.2</span> Randomness from your computer</h2>
<p>Computers make it easy to generate random numbers for resampling.</p>
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What do we mean by random?
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<p>Random numbers are numbers where it is impossible to predict which number is coming next. If we ask the computer for a number between 0 and 9, we will get one of the numbers 0 though 9, but we cannot do any better than that in predicting <em>which</em> number it will give us. There is an equal (10%) chance we will get any of the numbers 0 through 9 — just as there is when we roll a fair 10-sided die. We will go into more detail about what exactly we mean by <em>random</em> and <em>chance</em> later in the book (<a href="what_is_probability.html#sec-what-is-chance" class="quarto-xref"><span>Section 3.8</span></a>).</p>
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<p>We can use random numbers from computers to simulate our problem. For example, we can ask the computer to choose a random number between 0 and 9 to represent one ambulance. Let’s say the number 9 represents “out-of-order” and 0 through 8 “in operation”, then any one random number gives us a trial observation for a single ambulance. To get an experimental trial for a single day we look at 16 numbers and count how many of them are 9. We then look at, say, one hundred sets of 16 numbers and count the proportion of sets whose 16 numbers show three or more ambulances being “out-of-order”. Once again, that proportion estimates the probability that three or more ambulances will be out-of-order on any given day.</p>
<p>Soon we will do all these steps with some R code, but for now, consider Table <a href="#tbl-veh-numbers" class="quarto-xref">Table <span>2.1</span></a>. In each row, we placed 16 numbers, each one representing an ambulance. We added 25 such rows, each representing a simulation of one day.</p>
<div class="cell" data-layout-align="center">
<div id="tbl-veh-numbers" class="cell quarto-float quarto-figure quarto-figure-center anchored" data-layout-align="center">
<figure class="quarto-float quarto-float-tbl figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-veh-numbers-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 2.1: 25 simulations of 16 ambulances
</figcaption>
<div aria-describedby="tbl-veh-numbers-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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<table class="lightable-paper lightable-striped lightable-hover do-not-create-environment cell caption-top table table-sm table-striped small" data-quarto-postprocess="true">
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<tr class="header">
<th style="text-align: left;" data-quarto-table-cell-role="th"></th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A1</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A2</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A3</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A4</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A5</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A6</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A7</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A8</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A9</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A10</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A11</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A12</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A13</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A14</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A15</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">A16</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">Day 1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">1</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 4</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">6</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 5</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 6</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">5</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">7</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">7</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 10</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">1</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 11</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 12</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">9</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 13</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 14</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 15</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 16</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">8</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 17</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">4</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 18</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">1</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 19</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 20</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">6</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 21</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 22</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">2</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 23</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">0</td>
</tr>
<tr class="even">
<td style="text-align: left;">Day 24</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Day 25</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">4</td>