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201 lines
8.2 KiB
HTML
201 lines
8.2 KiB
HTML
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<title>Probability: Complement</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Probability: Complement</h1>
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<p class="center"><span class="larger">Complement of an Event: All outcomes that are <b>NOT</b> the event</span>. </p>
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<table cellspacing="10" border="0">
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<tbody>
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<tr>
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<td align="center"><img src="images/head-tails.jpg" alt="pair of dice" height="70" width="81"></td>
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<td>When the event is <b>Heads</b>, the complement is <b>Tails</b></td>
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</tr>
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<tr>
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<td><img src="images/month-thumb.gif" alt="weeks" height="58" width="100"></td>
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<td>When the event is <b>{Monday, Wednesday}</b> the complement is <b>{Tuesday, Thursday, Friday, Saturday, Sunday}</b></td>
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</tr>
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<tr>
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<td><img src="images/cards.jpg" alt="cards" height="78" width="100"></td>
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<td>When the event is <b>{Hearts}</b> the complement is <b>{Spades, Clubs, Diamonds, Jokers}</b></td>
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</tr>
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</tbody></table>
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<p>So the Complement of an event is all the <b>other</b> outcomes (<b>not</b> the ones we want).</p>
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<p>And together the Event and its Complement make all possible outcomes.</p>
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<h2>Probability</h2>
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<p class="center larger">Probability of an event happening = <span class="intbl"><em>Number of ways it can happen</em><strong>Total number of outcomes</strong></span> </p>
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<div class="example">
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<h3>Example: the chances of rolling a "4" with a die</h3>
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<p><b>Number of ways it can happen: 1</b> (there is only 1 face with a "4" on it)</p>
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<p><b>Total number of outcomes: 6</b> (there are 6 faces altogether)</p>
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<p class="center">So the probability = <span class="intbl"><em>1</em><strong>6</strong></span></p>
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</div>
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<p>The probability of an event is shown using "P":</p>
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<p class="center larger"><b>P(A)</b> means "Probability of Event A"</p>
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<p>The complement is shown by a little mark after the letter such as <span class="larger">A'</span> (or sometimes <span class="larger">A<sup>c</sup></span> or <span class="larger" style="text-decoration:overline">A</span>):</p>
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<p class="center larger"><b>P(A')</b> means "Probability of the complement of Event A"</p>
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<p>The two probabilities always add to 1</p>
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<p class="center larger">P(A) + P(A') = 1</p>
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<div class="example">
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<h3>Example: Rolling a "5" or "6" </h3>
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<p style="float:right; margin: 10px;"><img src="images/probability-complement.svg" alt="probability complement"></p>
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<p><b>Event A</b> is {5, 6}</p>
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<p>Number of ways it can happen: 2</p>
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<p>Total number of outcomes: 6</p>
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<p class="center larger">P(A) = <span class="intbl"><em>2</em><strong>6</strong></span> = <span class="intbl"><em>1</em><strong>3</strong></span></p>
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<p> </p>
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<p>The <b>Complement of Event A</b> is {1, 2, 3, 4} </p>
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<p>Number of ways it can happen: 4</p>
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<p>Total number of outcomes: 6</p>
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<p class="center larger">P(A') = <span class="intbl">
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<em>4</em>
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<strong>6</strong>
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</span> = <span class="intbl">
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<em>2</em>
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<strong>3</strong>
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</span></p>
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<p>Let us add them:</p>
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<p class="center larger">P(A) + P(A') = <span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> + <span class="intbl">
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<em>2</em>
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<strong>3</strong>
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</span> = <span class="intbl">
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<em>3</em>
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<strong>3</strong>
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</span> = 1</p>
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<p>Yep, that makes 1</p>
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<p>It makes sense, right? <b>Event A</b> plus all outcomes that are <b>not Event A</b> make up all possible outcomes.</p>
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</div>
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<p> </p>
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<div>
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<h2>Why is the Complement Useful?</h2>
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<p> It is sometimes easier to work out the complement first.</p>
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</div>
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<div class="example">
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<p style="float:right; margin: 5px;"><img src="../geometry/images/pair-dice2.svg" alt="pair of dice"></p>
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<h3>Example. Throw two dice. What is the probability the two scores are <b>different</b>?</h3>
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<p>Different scores are like getting a <b>2 and 3</b>, or a <b>6 and 1</b>. It is a long list:</p>
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<p class="center larger">A = { (1,2), (1,3), (1,4), (1,5), (1,6), <br>
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(2,1), (2,3), (2,4), (1,5), (1,6),<br>
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(3,1), (3,2), ... etc ! } </p>
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<p> </p>
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<p>But the complement (which is when the two scores are the same) is only <b>6 outcomes</b>:</p>
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<p class="center larger">A' = { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) }</p>
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<p>And its probability is:</p>
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<p class="center larger">P(A') = <span class="intbl"><em>6</em><strong>36</strong></span> = <span class="intbl"><em>1</em><strong>6</strong></span></p>
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<!-- P(A') = 6/36 = 1/6 -->
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<p> </p>
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<p>Knowing that P(A) and P(A') together make 1, we can calculate:</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td><span class="large">P(A)</span> </td>
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<td><span class="large">= 1 − P(A') </span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= 1 − <span class="intbl"><em>1</em><strong>6</strong></span></span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <span class="intbl"><em>5</em><strong>6</strong></span> </span></td>
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</tr>
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</tbody></table>
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<!-- P(A) = 1 - P(A') = 1 - 1/6 = 5/6 -->
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<p><br>
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So in this case (and many others) it is easier to work out <b>P(A')</b> first, then calculate <b>P(A) = 1 − P(A')</b></p>
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</div>
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<p> </p>
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<div class="questions">
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<script>getQ(3082, 3083, 3084, 3085, 3086, 3087, 3821, 3822, 3823, 184);</script>
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</div>
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<div class="related"><a href="probability.html">Probability</a> <a href="index.html">Data Index</a>
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