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      <text>35.2.2.6 Utility of the complement</text>
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      <text><![CDATA[<p>The use of the event \(A'\) can sometimes simplify the calculation of the probability \(\text{P}(A)\). For example, suppose that two 6-faced dice are thrown and we require the probability of the event\[A\text{: that we obtain a total score of at least four.}\]There are many combinations that produce a total score of at least four; however there are only \(3\) combinations that produce a total score of two or three which is the complementary event to the one of interest. The event \(A'=\{(1,1),(1,2),(2,1)\}\) (where we use an obvious notation of stating the total score on the first die followed by the score on the second die) is the complement of \(A\).</p>
<p> Now \(\text{P}(A')=\frac{3}{36}\) since there are \(6\times 6\) possible combinations in throwing two dice. Thus \[\qquad \text{P}(A)=1-\text{P}(A')=1-\frac{3}{36}=\frac{33}{36}=\frac{11}{12}.\]</p>]]></text>
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