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    <name>
      <text>35.2.1.9 Complement</text>
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      <text><![CDATA[<h4> Complement </h4>
<p> We have met the complement before in relation to sets. We consider it again
here in relation to sample spaces and events. The complement of an event is the set of outcomes
which are not members of the event.</p>
<p> For example, the experiment of throwing a 6-faced die has sample space \(S = \{1,2,3,4,5,6\}\).</p>
<ul>
<li> The event “score of 3 or more is obtained” is the set \(\{3,4,5,6\}\). The complement of this event is \(\{1,2\}\) which can be described in words as “score of 3 or more is not obtained” or “score of 1 or 2 is obtained”. </li>
<li> The event “even score is obtained” is the set \(\{2,4,6\}\). The complement of this event is \(\{1,3,5\}\) or, in words “even score is not obtained” or “odd score is obtained”. </li>
</ul>
<p> In a previous exercise concerning tossing three coins:</p>
<ul>
<li> the complement of event (a) is \(\{TTT,THH,HTH,HHT,HHH\}\). </li>
<li> the complement of event (b) is \(\{THH,HTH,HHT,HHH\}\). </li>
<li> the complement of event (c) is \(\{TTT\}\). </li>
</ul>
<p> <strong>Notation:</strong> It is customary to use a capital letter to denote an event. For example, \(A = \{\)two Heads are thrown\(\}\). The complementary event is denoted \(A'\).</p>
<hr>
<h4 class="HELM_example"> Example </h4>
<p>State, in words, what are the complements of each of the following events in
relation to the experiment of throwing three coins (avoid using the word not):</p>
<ol class="HELM_parts_inline">
<li> two Heads. </li>
<li> at least two Heads. </li>
<li> at most two Heads. </li>
</ol>
<h4 class="HELM_solution"> Solution </h4>
<ol class="HELM_parts">
<li> no Heads, one Head or three Heads. </li>
<li> no Heads or one Head (or equivalently, at most one Head).</li>
<li> three Heads. </li>
</ol>]]></text>
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