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<h1 class="center">Mutually Exclusive Events</h1>
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<p><img src="images/road-fork.jpg" alt="road fork" width="100%"></p>
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<p class="words"><b>Mutually Exclusive</b>: can't happen at the same time. </p>
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<p>Examples:</p>
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<ul>
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<li>Turning left and turning right are Mutually Exclusive (you can't do both at the same time)</li>
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<li>Tossing a coin: Heads and Tails are Mutually Exclusive</li>
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<li>Cards: Kings and Aces are Mutually Exclusive</li>
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</ul>
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<p>What is <b>not</b> Mutually Exclusive:</p>
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<ul>
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<li>Turning left and scratching your head can happen at the same time</li>
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<li>Kings and Hearts, because we can have a King of Hearts! </li>
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</ul>
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<p>Like here: </p>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr align="center">
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<td><img src="images/set-aces-kings.svg" alt="set aces kings separate"></td>
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<td> </td>
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<td><img src="images/set-hearts-kings.svg" alt="set hearts kings joins at king of hearts"></td>
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</tr>
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<tr align="center">
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<td>Aces and Kings are <br>
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<b>Mutually Exclusive</b><br>
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(can't be both)</td>
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<td> </td>
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<td>Hearts and Kings are <b><br>
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not</b> Mutually Exclusive <br>
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(can be both)</td>
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</tr>
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</tbody></table>
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</div>
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<h2>Probability </h2>
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<p>Let's look at the probabilities of Mutually Exclusive events. But first, a definition:</p>
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<div class="def">
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<p class="center larger">Probability of an event happening = <span class="intbl">
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<em>Number of ways it can happen</em>
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<strong>Total number of outcomes</strong>
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</span></p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: there are 4 Kings in a deck of 52 cards. What is the probability of picking a King?</h3>
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<p><b>Number of ways it can happen: 4</b> (there are 4 Kings)</p>
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<p><b>Total number of outcomes: 52</b> (there are 52 cards in total)</p>
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<p class="center larger">So the probability = <span class="intbl">
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<em>4</em>
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<strong>52</strong>
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</span> = <span class="intbl">
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<em>1</em>
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<strong>13</strong>
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</span></p>
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</div>
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<h2>Mutually Exclusive</h2>
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<p>When two events (call them "A" and "B") are Mutually Exclusive it is <b>impossible</b> for them to happen together:</p>
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<p class="center larger"><b>P(A and B) = 0</b> </p>
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<p class="center"><i>"The probability of A and B together equals 0 (impossible)"</i></p>
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<div class="example">
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<h3>Example: King AND Queen</h3>
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<p>A card cannot be a King AND a Queen at the same time! </p>
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<ul>
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<li>The probability of a King <b>and</b> a Queen is <b>0</b> (Impossible)</li>
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</ul>
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</div>
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<p class="center"> </p>
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<p>But, for Mutually Exclusive events, the probability of A <b>or</b> B is the sum of the individual probabilities:</p>
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<p class="center larger"><b>P(A or B) = P(A) + P(B)</b></p>
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<p class="center"><i>"The probability of A <b>or</b> B equals the probability of A <b>plus</b> the probability of B"</i></p>
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<div class="example">
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<h3>Example: King OR Queen</h3>
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<p>In a Deck of 52 Cards:</p>
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<ul>
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<li>the probability of a King is 1/13, so <b>P(King)=1/13</b></li>
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<li>the probability of a Queen is also 1/13, so <b>P(Queen)=1/13</b></li>
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</ul>
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<p> </p>
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<p>When we combine those two Events:</p>
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<ul>
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<li>The probability of a King <b>or</b> a Queen is (1/13) + (1/13) = <b>2/13</b></li>
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</ul>
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<p>Which is written like this: </p>
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<p class="center larger">P(King or Queen) = (1/13) + (1/13) = 2/13</p>
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</div>
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<p>So, we have: </p>
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<ul>
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<li>P(King and Queen) = 0</li>
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<li>P(King or Queen) = (1/13) + (1/13) = 2/13</li>
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</ul>
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<h2>Special Notation</h2>
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<p>Instead of "and" you will often see the symbol <b class="large">∩</b> (which is the "Intersection" symbol used in <a href="../sets/venn-diagrams.html">Venn Diagrams</a>)</p>
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<p>Instead of "or" you will often see the symbol <span class="large"><b>∪</b></span> (the "Union" symbol)</p>
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<p>So we can also write: </p>
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<ul>
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<li>P(King <b class="large">∩</b> Queen) = 0</li>
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<li>P(King <span class="large"><b>∪</b></span> Queen) = (1/13) + (1/13) = 2/13</li>
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</ul>
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<div class="example">
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<p style="float:right; margin: 10px;"><img src="../images/soccer-teams.jpg" alt="soccer teams" height="86" width="209"></p>
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<h3>Example: Scoring Goals</h3>
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<p>If the probability of:</p>
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<ul>
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<li>scoring no goals (Event "A") is <b>20%</b></li>
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<li>scoring exactly 1 goal (Event "B") is <b>15% </b></li>
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</ul>
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<p>Then:</p>
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<ul>
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<li>The probability of scoring no goals <b>and</b> 1 goal is <b>0</b> (Impossible)</li>
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<li>The probability of scoring no goals <b>or</b> 1 goal is 20% + 15% = <b>35%</b></li>
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</ul>
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<p> </p>
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<p>Which is written: </p>
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<p class="center larger">P(A <b>∩</b> B) = 0 </p>
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<p class="center larger">P(A <b>∪</b> B) = 20% + 15% = 35%</p>
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</div>
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<h2>Remembering</h2>
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<p>To help you remember, think:</p>
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<div class="center80">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/union-cup.jpg" alt="union cup" height="88" width="102"></p>
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<p class="center"><b>"Or</b> has <b>more</b> ... <b>than And</b>"</p>
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<p class="center">Also <b>∪</b> is like a cup which holds <b>more</b> than <b>∩</b></p>
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</div>
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<h2>Not Mutually Exclusive</h2>
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<p>Now let's see what happens when events are <b>not Mutually Exclusive</b>.</p>
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<h3>Example: Hearts and Kings</h3>
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<p class="center"><img src="images/set-hearts-kings.svg" alt="set hearts kings joins at king of hearts"></p>
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<table>
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<tbody>
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<tr>
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<td>
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<p>Hearts <b>and</b> Kings together is only the King of Hearts:</p></td>
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<td><img src="images/set-hearts-kings-union.svg" alt="set hearts kings union"></td>
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</tr>
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</tbody></table>
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<p>But Hearts <b>or</b> Kings is:</p>
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<ul>
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<li>all the Hearts (13 of them)</li>
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<li>all the Kings (4 of them)</li>
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</ul>
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<p><b>But that counts the King of Hearts twice! </b></p>
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<p class="center">So we correct our answer, by subtracting the extra "and" part:</p>
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<p class="center"><img src="images/set-hearts-kings-sum.svg" alt="set hearts kings sum"></p>
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<p class="center larger">16 Cards = 13 Hearts + 4 Kings − the 1 extra King of Hearts</p>
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<p class="center">Count them to make sure this works!</p>
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<p>As a formula this is:</p>
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<p class="center larger"><b>P(A or B) = P(A) + P(B) − P(A and B)</b></p>
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<p class="center"><i>"The probability of A <b>or</b> B equals
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the probability of A <b>plus</b> the probability of B <br>
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<b>minus</b> the probability of A <b>and</b> B"</i></p>
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<p>Here is the <b>same formula</b>, but using <b>∪</b> and <b>∩</b>:</p>
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<p class="center larger"><b>P(A ∪ B) = P(A) + P(B) − P(A ∩ B)</b></p>
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<h2>A Final Example</h2>
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<p class="larger"> 16 people study French, 21 study Spanish and there are 30 altogether. Work out the probabilities!</p>
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<p>This is definitely a case of <b>not</b> Mutually Exclusive (you can study French AND Spanish).</p>
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<p>Let's say <span class="larger"><b>b</b></span> is how many study both languages:</p>
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<ul>
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<li>people studying French Only must be <span class="larger">16-b</span></li>
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<li>people studying Spanish Only must be <span class="larger">21-b</span></li>
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</ul>
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<p>And we get:</p>
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<p class="center"><img src="images/set-language-ex1.svg" alt="set language ex1"></p>
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<p>And we know there are <b>30</b> people, so:</p>
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<div class="so"> (16−b) + b + (21−b) = 30<br>
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</div>
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<div class="so"> 37 − b = 30<br>
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</div>
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<div class="so">b = 7</div>
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<p>And we can put in the correct numbers:</p>
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<p class="center"><img src="images/set-language-ex2.svg" alt="set language ex2"></p>
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<p>So we know all this now:</p>
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<ul>
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<li>P(French) = 16/30<br>
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</li>
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<li>P(Spanish) = 21/30<br>
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</li>
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<li>P(French Only) = 9/30<br>
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</li>
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<li>P(Spanish Only) = 14/30<br>
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</li>
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<li>P(French or Spanish) = 30/30 = 1<br>
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</li>
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<li>P(French and Spanish) = 7/30</li>
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</ul>
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<p>Lastly, let's check with our formula:</p>
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<p class="center larger"><b>P(A or B) = P(A) + P(B) − P(A and B)</b></p>
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<p>Put the values in:</p>
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<p class="center larger"><b>30/30 = 16/30 + 21/30 − 7/30</b></p>
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<p class="center">Yes, it works!</p>
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<p> </p>
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<h2>Summary:</h2>
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<h3>Mutually Exclusive</h3>
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<ul>
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<div class="bigul">
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<li>A <b>and</b> B together is impossible: <b>P(A and B) = 0</b></li>
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<li>A <b>or</b> B is the sum of A and B: <b>P(A or B) = P(A) + P(B)<br>
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</b></li>
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</div>
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</ul>
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<ul>
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<div class="bigul"></div>
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</ul>
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<h3>Not Mutually Exclusive</h3>
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<div class="bigul">
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<ul>
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<li>A <b>or</b> B is the sum of A and B minus A <b>and</b> B: <b>P(A or B) = P(A) + P(B) − P(A and B)</b></li>
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</ul>
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</div>
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<h3>Symbols</h3>
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<div class="bigul">
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<ul>
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<li><b>And</b> is <b class="large">∩</b> (the "Intersection" symbol)</li>
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<li><b>Or</b> is <span class="large"><b>∪</b></span> (the "Union" symbol) </li>
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</ul>
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</div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(3101, 3102, 3103, 3104, 3105, 3106, 3107, 3834, 3835, 3836);</script>
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</div>
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<div class="related"><a href="probability.html">Probability</a>
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<a href="index.html">Data Index</a>
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<a href="probability-events-types.html">Types of Events</a></div>
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