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      <text>35.4.1.1 The theorem of total probability</text>
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      <text><![CDATA[<h3> 1.1 The theorem of Total Probability and Bayes' theorem </h3>
<p> To establish these results we start with the definition of a partition of a sample space. </p>
<h4> A partition of a sample space </h4>
<p> The collection of events \(A_1,A_2,\dots,A_n\) is said to <strong>partition</strong> a sample space \(S\) if </p>
<ol type="a">
<li> \(A_1\cup A_2 \cup\cdots\cup A_n = S\); </li>
<li> \(A_i\cap A_j = \varnothing\) for all \(i, j\) with \(i\neq j\); </li>
<li> \(A_i\neq\varnothing\) for all \(i\). </li>
</ol>
<p> In essence, a partition is a collection of non-empty, non-overlapping subsets of a sample space whose union is the sample space itself. The definition is illustrated by the figure below. </p>
<p style="text-align: center;"><strong><img src="@@PLUGINFILE@@/3-5-4-partition1.svg" alt="" role="presentation"><br></strong></p><p style="text-align: center;"><strong>Figure:</strong> Partition of a set.</p>
<p> If \(B\) is any event within \(S\) then we can express \(B\) as the union of subsets:\[B = (B \cap A_1) \cup (B \cap A_2) \cup\cdots\cup(B \cap A_n)\]The definition is illustrated in the figure below in which an event B in S is represented by the shaded region.</p><p style="text-align: center;"><strong><img src="@@PLUGINFILE@@/3-5-4-partition2.svg" alt="" role="presentation"><br></strong></p>
<p style="text-align: center;"><strong>Figure:</strong> A subset in a set partition.</p>
<p> The bracketed events \((B\cap A_1),(B\cap A_2),\dots,(B\cap A_n)\) are mutually exclusive (if one occurs then none of the others can occur) and so, using the addition law of probability for mutually exclusive events:\[\mathbb{P}(B) = \mathbb{P}(B\cap A_1) + \mathbb{P}(B\cap A_2) +\cdots+ \mathbb{P}(B\cap A_n)\]Each of the probabilities on the right-hand side may be expressed in terms of conditional probabilities:\[\mathbb{P}(B\cap A_i) = \mathbb{P}(B|A_i)\mathbb{P}(A_i)\quad\text{for all }i.\]Using these in the expression for \(\mathbb{P}(B)\), above, gives:\[\begin{aligned}\mathbb{P}(B) &amp;= \mathbb{P}(B|A_1)\mathbb{P}(A_1) + \mathbb{P}(B|A_2)\mathbb{P}(A_2) +\cdots+ \mathbb{P}(B|A_n)\mathbb{P}(A_n)\\ &amp;= \sum_{i=1}^n\mathbb{P}(B|A_i)\mathbb{P}(A_i).\end{aligned}\]This is the <strong>theorem of Total Probability</strong>. A related theorem with many applications in statistics can be deduced from this, known as <strong>Bayes’ theorem</strong>.</p>]]></text>
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