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292 lines
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292 lines
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<h1 class="center">Conditional Probability</h1>
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<p class="center"><i>How to handle <b>Dependent Events</b></i> </p>
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<p>Life is full of random events! You need to get a "feel" for them to be a smart and successful person. </p>
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<h2>Independent Events </h2>
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<p> Events can be "<a href="probability-events-independent.html">Independent</a>", meaning each event is <b>not affected</b> by any other events.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/head-tails-dollar.jpg" width="102" height="96" alt="head tails coin" /></p>
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<h3>Example: Tossing a coin.</h3>
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<p>Each toss of a coin is a perfect isolated thing. </p>
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<p>What it did in the past will not affect the current toss.</p>
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<p>The chance is simply 1-in-2, or 50%, just like ANY toss of the coin.</p>
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<p>So each toss is an <b>Independent Event</b>.</p>
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</div>
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<h2>Dependent Events</h2>
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<p> But events can also be "dependent" ... which means they <b>can be affected by previous events</b> ... </p>
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<div class="example"><img style="float:right; margin: 10px;" src="images/probability-marbles1.svg" alt="probability marbles" />
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<h3>Example: Marbles in a Bag</h3>
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<p> 2 blue and 3 red marbles are in a bag. </p>
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<p>What are the chances of getting a blue marble?</p>
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<p class="center">The chance is <b>2 in 5</b></p>
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<p><b>But after taking one out</b> the chances change!</p>
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<p>So the next time:</p>
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<p class="center"><img src="images/probability-marbles2.svg" alt="probability marbles" /><br>
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if we got a <b>red</b> marble before, then the chance of a blue marble next is <b>2 in 4</b></p>
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<p class="center"><img src="images/probability-marbles3.svg" alt="probability marbles" /><br>
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if we got a <b>blue</b> marble before, then the chance of a blue marble next is <b>1 in 4</b></p>
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</div>
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<p>This is because we are <b>removing</b> marbles from the bag.</p>
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<p>So the next event <b>depends on</b> what happened in the previous event, and is called <b>dependent</b>. </p>
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<div class="center80">
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<p><span class="larger">Replacement</span></p>
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<p>Note: if we <b>replace</b> the marbles in the bag each time, then the chances do <b>not</b> change and the events are <a href="probability-events-independent.html">independent</a>:</p>
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<ul>
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<li><b>With</b> Replacement: the events are <b>Independent</b> (the chances don't change)</li>
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<li><b>Without</b> Replacement: the events are <b>Dependent</b> (the chances change)</li>
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</ul>
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</div>
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<p><b>Dependent</b> events are what we look at here.</p>
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<h2>Tree Diagram</h2>
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<p>A <a href="probability-tree-diagrams.html">Tree Diagram</a>: is a wonderful way to picture what is going on, so let's build one for our marbles example.</p>
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<p>There is a 2/5 chance of pulling out a Blue marble, and a 3/5 chance for Red:</p>
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<p><img src="images/probability-marbles-tree1.svg" alt="probability marbles tree 1" /></p>
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<p>We can go one step further and see what happens when we pick a second marble:</p>
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<p><img src="images/probability-marbles-tree2.svg" alt="probability marbles tree 2" /></p>
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<p>If a blue marble was selected first there is now a 1/4 chance of getting a blue marble and a 3/4 chance of getting a red marble.</p>
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<p>If a red marble was selected first there is now a 2/4 chance of getting a blue marble and a 2/4 chance of getting a red marble.</p>
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<div class="center80">
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<p>Now we can answer questions like <b>"What are the chances of drawing 2 blue marbles?"</b></p>
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<p>Answer: it is a <b>2/5 chance</b> followed by a <b>1/4 chance</b>:</p>
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<p class="center"><img src="images/probability-marbles-tree3.svg" style="max-width:100%" alt="probability marbles tree 3" /></p>
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<p>Did you see how we multiplied the chances? And got 1/10 as a result.</p>
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<p><b>The chances of drawing 2 blue marbles is 1/10</b></p>
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</div>
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<h2>Notation</h2>
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<p><b>We love notation in mathematics!</b> It means we can then use the power of algebra to play around with the ideas. So here is the notation for probability:</p>
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<p class="center larger">P(A) means "Probability Of Event A"</p>
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<p>In our marbles example Event A is "get a Blue Marble first" with a probability of 2/5:</p>
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<p class="center"><b>P(A) = 2/5</b></p>
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<p>And Event B is "get a Blue Marble second" ... but for that we have 2 choices:</p>
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<ul>
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<li>If we got a <b>Blue Marble first</b> the chance is now <b>1/4 </b></li>
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<li>If we got a <b>Red Marble first</b> the chance is now <b>2/4</b></li>
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</ul>
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<p>So we have to say <b>which one we want</b>, and use the symbol "|" to mean "given":</p>
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<p class="center larger">P(B|A) means "Event B <b>given</b> Event A"</p>
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<p class="center">In other words, event A has already happened, now what is the chance of event B?</p>
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<p><span class="center larger">P(B|A)</span> is also called the "Conditional Probability" of B given A.</p>
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<p>And in our case:</p>
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<p class="center"> <b>P(B|A) = 1/4</b></p>
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<p>So the probability of getting <b>2 blue marbles</b> is:</p>
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<p class="center"><img src="images/probability-marbles-tree4.svg" style="max-width:100%" alt="probability marbles tree 4" /></p>
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<p>And we write it as</p>
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<p class="center"><img src="images/probability-independent-formula1.svg" alt="P( A and B ) = P(A) times P(B given A)" /></p>
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<p class="center"><i>"Probability of <b>event A and event B</b> equals <br />
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the probability of <b>event A</b> times the probability of <b>event B given event A</b>"</i></p>
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<p> </p>
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<p>Let's do the next example using only notation:</p>
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<div class="example">
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<h3>Example: Drawing 2 Kings from a Deck </h3>
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<p><b>Event A</b> is drawing a King first, and <b>Event B</b> is drawing a King second.</p>
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<p>For the first card the chance of drawing a King is 4 out of 52 (there are 4 Kings in a deck of 52 cards):</p>
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<p class="center larger">P(A) = 4/52</p>
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<p>But after removing a King from the deck the probability of the 2nd card drawn is <b>less</b> likely to be a King (only 3 of the 51 cards left are Kings):</p>
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<p class="center larger">P(B|A) = 3/51</p>
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<p>And so:</p>
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<p class="center larger"><b>P(A and B) = P(A) x P(B|A)</b> = (4/52) x (3/51) = 12/2652 = <b>1/221</b></p>
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<p>So the chance of getting 2 Kings is 1 in 221, or about 0.5%</p>
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</div>
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<h2>Finding Hidden Data</h2>
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<p>Using Algebra we can also "change the subject" of the formula, like this:</p>
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<table border="0" align="center">
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<tr>
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<td align="right">Start with: </td>
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<td> </td>
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<td class="larger">P(A and B) = P(A) x P(B|A)</td>
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</tr>
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<tr>
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<td align="right">Swap sides:</td>
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<td> </td>
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<td class="larger"> P(A) x P(B|A) = P(A and B) </td>
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</tr>
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<tr>
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<td align="right">Divide by P(A):</td>
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<td> </td>
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<td class="larger">P(B|A) = P(A and B) / P(A)</td>
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</tr>
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</table>
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<p>And we have another useful formula:</p>
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<p class="center"><img src="images/probability-independent-formula2.gif" width="286" height="64" alt="P(B given A) = P( A and B ) / P(A) " /></p>
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<p class="center"><i>"The probability of <b>event B given event A</b> equals <br />
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the probability of <b>event A and event B</b> divided by the probability of <b>event A</b></i></p>
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<div class="example">
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<h3>Example: Ice Cream</h3>
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<p>70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberry. </p>
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<p>What percent of those who like Chocolate also like Strawberry?</p>
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<p class="larger">P(Strawberry|Chocolate) = P(Chocolate and Strawberry) / P(Chocolate)</p>
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<div class="so"> 0.35 / 0.7 = 50% </div>
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<p>50% of your friends who like Chocolate also like Strawberry</p>
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</div>
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<h2> </h2>
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<p style="float:right; margin: 10px;"><img src="../images/soccer-teams.jpg" width="209" height="86" alt="soccer teams" /></p>
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<h2>Big Example: Soccer Game</h2>
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<p>You are off to soccer, and want to be the Goalkeeper, but that depends who is the Coach today:</p>
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<ul>
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<li>with Coach Sam the probability of being Goalkeeper is <b>0.5</b></li>
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<li>with Coach Alex the probability of being Goalkeeper is <b>0.3</b></li>
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</ul>
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<p>Sam is Coach more often ... about 6 out of every 10 games (a probability of <b>0.6</b>).</p>
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<p class="larger">So, what is the probability you will be a Goalkeeper today?</p>
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<p> </p>
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<p>Let's build a <a href="probability-tree-diagrams.html">tree diagram</a>. First we show the two possible coaches: Sam or Alex:</p>
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<p class="center"><img src="images/tree-diagram-ex1-1.svg" alt="tree diagram 1" /></p>
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<p>The probability of getting Sam is 0.6, so the probability of Alex must be 0.4 (together the probability is 1)</p>
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<p>Now, if you get Sam, there is 0.5 probability of being Goalie (and 0.5 of not being Goalie):</p>
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<p class="center"><img src="images/tree-diagram-ex1-2.svg" alt="tree diagram 2" /></p>
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<p>If you get Alex, there is 0.3 probability of being Goalie (and 0.7 not):</p>
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<p class="center"><img src="images/tree-diagram-ex1-3.svg" alt="tree diagram 3" /></p>
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<p>The tree diagram is complete, now let's calculate the overall probabilities. Remember that:</p>
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<p class="center larger">P(A and B) = P(A) x P(B|A)</p>
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<p>Here is how to do it for the "Sam, Yes" branch:</p>
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<p class="center"><img src="images/tree-diagram-ex1-4.svg" alt="tree diagram 4" style="max-width:100%" /></p>
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<p>(When we take the 0.6 chance of Sam being coach times the 0.5 chance that Sam will let you be Goalkeeper we end up with an 0.3 chance.)</p>
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<p>But we are not done yet! We haven't included Alex as Coach:</p>
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<p class="center"><img src="images/tree-diagram-ex1-5.svg" alt="tree diagram 5" style="max-width:100%" /></p>
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<p>An 0.4 chance of Alex as Coach, followed by an 0.3 chance gives 0.12</p>
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<p><b>And the two "Yes" branches of the tree together make:</b></p>
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<p class="center larger">0.3 + 0.12 = <b>0.42 probability</b> of being a Goalkeeper today</p>
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<p class="center">(That is a 42% chance)</p>
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<h3>Check</h3>
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<p>One final step: complete the calculations and make sure they add to 1:</p>
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<p class="center"><img src="images/tree-diagram-ex1-6.svg" alt="tree diagram 6" style="max-width:100%" /></p>
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<p class="larger center">0.3 + 0.3 + 0.12 + 0.28 = 1</p>
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<p class="center">Yes, they add to <b>1</b>, so that looks right.</p>
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<h2>Friends and Random Numbers</h2>
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<p>Here is another quite different example of Conditional Probability. </p>
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<p class="larger"><b>4 friends</b> (Alex, Blake, Chris and Dusty) each choose a random number between 1 and 5. What is the chance that any of them chose the same number?</p>
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<p>Let's add our friends one at a time ...</p>
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<p> </p>
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<h3>First, what is the chance that Alex and Blake have the same number?</h3>
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<p>Blake compares his number to Alex's number. There is a 1 in 5 chance of a match.</p>
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<p>As a <a href="probability-tree-diagrams.html">tree diagram</a>:</p>
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<p class="center"><img src="images/events-dependent-ex-1.svg" alt="events dependent 1" /></p>
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<p class="center">Note: "Yes" and "No" together makes 1 <br />
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(1/5 + 4/5 = 5/5 = 1)</p>
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<h3> </h3>
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<h3>Now, let's include Chris ... </h3>
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<p>But there are now two cases to consider:</p>
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<ul>
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<li>If Alex and Blake <b>did</b> match, then Chris has only <b>one number</b> to compare to.</li>
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<li>But if Alex and Blake <b>did not</b> match then Chris has <b>two numbers</b> to compare to.</li>
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</ul>
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<p>And we get this:</p>
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<p class="center"><img src="images/events-dependent-ex-2.svg" alt="events dependent 2" style="max-width:100%" /></p>
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<p>For the top line (Alex and Blake <b>did</b> match) we already have a match (a chance of 1/5).</p>
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<p>But for the "Alex and Blake <b>did not</b> match" there is now a <b>2/5</b> chance of Chris matching (because Chris gets to match his number against both Alex and Blake). </p>
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<p>And we can work out the combined chance by <b>multiplying the chances</b> it took to get there:</p>
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<div class="dotpoint">
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<p>Following the "No, Yes" path ... there is a 4/5 chance of No, followed by a 2/5 chance of Yes:</p>
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<div class="so">(4/5) × (2/5) = 8/25</div>
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</div>
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<div class="dotpoint">
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<p>Following the "No, No" path ... there is a 4/5 chance of No, followed by a 3/5 chance of No:</p>
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<div class="so">(4/5) × (3/5) = 12/25</div>
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</div>
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<p>Also notice that when we add all chances together we still get 1 (a good check that we haven't made a mistake): </p>
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<p class="center">(5/25) + (8/25) + (12/25) = 25/25 = 1</p>
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<p> </p>
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<h3>Now what happens when we include Dusty?</h3>
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<p>It is the same idea, just more of it:</p>
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<p class="center"><img src="images/events-dependent-ex-3.svg" alt="events dependent 3" style="max-width:100%" /></p>
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<p>OK, that is all 4 friends, and the "Yes" chances together make 101/125:</p>
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<p class="center large">Answer: <b>101/125</b></p>
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<p> </p>
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<div class="center80">
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<p>But here is something interesting ... if we follow the "No" path we can <b>skip all the other calculations</b> and make our life easier:</p>
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<p class="center"><img src="images/events-dependent-ex-4.svg" alt="events dependent 4" style="max-width:100%" /></p>
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<p>The chances of <b>not matching</b> are:</p>
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<p class="center larger">(4/5) × (3/5) × (2/5) = <b>24/125</b></p>
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<p>So the chances of <b>matching</b> are:</p>
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<p class="center larger">1 - (24/125) = <b>101/125</b></p>
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<p>(And we didn't really need a tree diagram for that!)</p>
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</div>
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<p> </p>
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<p>And that is a popular trick in probability:</p>
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<p class="center larger">It is often easier to work out the "No" case<br>
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(and subtract from 1 for the "Yes" case)
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</p>
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<p>(This idea is shown in more detail at <a href="probability-shared-birthday.html"><span class="center">Shared Birthdays</span></a>.)</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(3088, 3089, 3090, 3091, 3092, 3093, 3094, 3095, 3827, 3828);</script>
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</div>
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<div class="related"><a href="probability.html">Probability</a> <a href="probability-tree-diagrams.html">Tree Diagrams</a>
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<a href="index.html">Data Index</a> </div>
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