<p>Let&nbsp;\(X\) be the sum of the outcomes obtained when&nbsp;\(2\) fair dice are thrown.</p>
<p>One constructs the possibility diagram that represents the outcomes of this experiment.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><caption>Possibility diagram for the sum of two dice (Die 1 vs Die 2)</caption><thead><tr><th style="text-align: left; --cell-justify: flex-start">Sum</th><th style="text-align: left; --cell-justify: flex-start">\(1\)</th><th style="text-align: left; --cell-justify: flex-start">\(2\)</th><th style="text-align: left; --cell-justify: flex-start">\(3\)</th><th style="text-align: left; --cell-justify: flex-start">\(4\)</th><th style="text-align: left; --cell-justify: flex-start">\(5\)</th><th style="text-align: left; --cell-justify: flex-start">\(6\)</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">\(1\)</td><td style="text-align: left; --cell-justify: flex-start">\(2\)</td><td style="text-align: left; --cell-justify: flex-start">\(3\)</td><td style="text-align: left; --cell-justify: flex-start">\(4\)</td><td style="text-align: left; --cell-justify: flex-start">\(5\)</td><td style="text-align: left; --cell-justify: flex-start">\(6\)</td><td style="text-align: left; --cell-justify: flex-start">\(7\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(2\)</td><td style="text-align: left; --cell-justify: flex-start">\(3\)</td><td style="text-align: left; --cell-justify: flex-start">\(4\)</td><td style="text-align: left; --cell-justify: flex-start">\(5\)</td><td style="text-align: left; --cell-justify: flex-start">\(6\)</td><td style="text-align: left; --cell-justify: flex-start">\(7\)</td><td style="text-align: left; --cell-justify: flex-start">\(8\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(3\)</td><td style="text-align: left; --cell-justify: flex-start">\(4\)</td><td style="text-align: left; --cell-justify: flex-start">\(5\)</td><td style="text-align: left; --cell-justify: flex-start">\(6\)</td><td style="text-align: left; --cell-justify: flex-start">\(7\)</td><td style="text-align: left; --cell-justify: flex-start">\(8\)</td><td style="text-align: left; --cell-justify: flex-start">\(9\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(4\)</td><td style="text-align: left; --cell-justify: flex-start">\(5\)</td><td style="text-align: left; --cell-justify: flex-start">\(6\)</td><td style="text-align: left; --cell-justify: flex-start">\(7\)</td><td style="text-align: left; --cell-justify: flex-start">\(8\)</td><td style="text-align: left; --cell-justify: flex-start">\(9\)</td><td style="text-align: left; --cell-justify: flex-start">\(10\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(5\)</td><td style="text-align: left; --cell-justify: flex-start">\(6\)</td><td style="text-align: left; --cell-justify: flex-start">\(7\)</td><td style="text-align: left; --cell-justify: flex-start">\(8\)</td><td style="text-align: left; --cell-justify: flex-start">\(9\)</td><td style="text-align: left; --cell-justify: flex-start">\(10\)</td><td style="text-align: left; --cell-justify: flex-start">\(11\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(6\)</td><td style="text-align: left; --cell-justify: flex-start">\(7\)</td><td style="text-align: left; --cell-justify: flex-start">\(8\)</td><td style="text-align: left; --cell-justify: flex-start">\(9\)</td><td style="text-align: left; --cell-justify: flex-start">\(10\)</td><td style="text-align: left; --cell-justify: flex-start">\(11\)</td><td style="text-align: left; --cell-justify: flex-start">\(12\)</td></tr></tbody></table></div></div>
<p>By counting the frequency of each occurrence of the sum, we make the following tabulation that shows the probability distribution of \(X\):</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start">\(x\)</th><th style="text-align: left; --cell-justify: flex-start">\(2\)</th><th style="text-align: left; --cell-justify: flex-start">\(3\)</th><th style="text-align: left; --cell-justify: flex-start">\(4\)</th><th style="text-align: left; --cell-justify: flex-start">\(5\)</th><th style="text-align: left; --cell-justify: flex-start">\(6\)</th><th style="text-align: left; --cell-justify: flex-start">\(7\)</th><th style="text-align: left; --cell-justify: flex-start">\(8\)</th><th style="text-align: left; --cell-justify: flex-start">\(9\)</th><th style="text-align: left; --cell-justify: flex-start">\(10\)</th><th style="text-align: left; --cell-justify: flex-start">\(11\)</th><th style="text-align: left; --cell-justify: flex-start">\(12\)</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">\(P(X = x)\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{1}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{2}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{3}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{4}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{5}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{6}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{5}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{4}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{3}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{2}{36}\)</td><td style="text-align: left; --cell-justify: flex-start">\(\frac{1}{36}\)</td></tr></tbody></table></div></div>
<p>It is straightforward to observe that the set of values this random variable can take is finite, and each of the probability associated to the values is in the interval \([0, 1]\). Additionally,</p>
<p style="text-align: center">\[\sum_{\text{all}\hspace{0.5em} x} P(X = x) = \frac{1}{36} + \frac{2}{36} + \frac{3}{36} + \frac{4}{36} + \frac{5}{36} + \frac{6}{36} + \frac{5}{36} + \frac{4}{36} + \frac{3}{36} + \frac{2}{36} + \frac{1}{36} = 1.\]</p>
<p>Given a probability density function of random variable \(X\) whose set of values is finite, we can use a table such as in Example \(2\) to display the probability distribution of \(X\). Another more visual representation is to use a histogram.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1777027108665-v41j8ls1azd.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

Let \(X\) be the sum of the outcomes obtained when \(2\) fair dice are thrown.

One constructs the possibility diagram that represents the outcomes of this experiment.

By counting the frequency of each occurrence of the sum, we make the following tabulation that shows the probability distribution of \(X\):

It is straightforward to observe that the set of values this random variable can take is finite, and each of the probability associated to the values is in the interval \([0, 1]\). Additionally,

\[\sum_{\text{all } x} P(X = x) = \frac{1}{36} + \frac{2}{36} + \frac{3}{36} + \frac{4}{36} + \frac{5}{36} + \frac{6}{36} + \frac{5}{36} + \frac{4}{36} + \frac{3}{36} + \frac{2}{36} + \frac{1}{36} = 1.\]

Given a probability density function of random variable \(X\) whose set of values is finite, we can use a table such as in Example \(2\) to display the probability distribution of \(X\). Another more visual representation is to use a histogram.

Discrete Random Variables (DRV) - General

Discrete Random Variables (DRV)

International Baccalaureate Higher Level Mathematics

Discrete Random Variables (DRV) - Discrete Random Variables (DRV) - General

Discrete Random Variables (DRV) - Discrete Random Variables (DRV) - General