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146 lines
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HTML
146 lines
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HTML
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<h1 class="center">Probability Tree Diagrams</h1>
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<p>Calculating probabilities can be hard, sometimes we add them, sometimes we multiply them, and often it is hard to figure out what to do ... <b>tree diagrams to the rescue!</b></p>
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<p> </p>
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<p>Here is a tree diagram for the toss of a coin:</p>
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<table border="0">
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<tr>
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<td><img src="images/probability-tree-coin1.svg" alt="probability branch, probability and outcome" /></td>
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<td> </td>
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<td valign="top"><p>There are two "branches" (Heads and Tails)</p>
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<ul>
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<li>The probability of each branch is written on the branch</li>
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<li>The outcome is written at the end of the branch</li>
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</ul> </td>
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</tr>
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</table>
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<p>We can extend the tree diagram to two tosses of a coin:</p>
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<p><img src="images/probability-tree-coin2.svg" alt="probability tree 2 tosses of coin" /></p>
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<p>How do we calculate the overall probabilities?</p>
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<ul>
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<li>We <b>multiply</b> probabilities <b>along the branches</b></li>
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<li>We <b>add</b> probabilities down <b>columns</b></li>
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</ul>
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<p><img src="images/probability-tree-coin3.svg" alt="probability tree calculations (multiply and add)" style="max-width:100%" /></p>
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<p>Now we can see such things as:</p>
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<ul>
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<li>The probability of "Head, Head" is 0.5×0.5 = <b>0.25</b></li>
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<li>All probabilities add to <b>1.0</b> (which is always a good check)</li>
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<li>The probability of getting at least one Head from two tosses is 0.25+0.25+0.25 = <b>0.75</b></li>
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<li>... and more</li>
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</ul>
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<p> </p>
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<p>That was a simple example using <a href="probability-events-independent.html">independent events</a> (each toss of a coin is independent of the previous toss), but tree diagrams are really wonderful for figuring out <a href="probability-events-conditional.html">dependent events</a> (where an event <b>depends on</b> what happens in the previous event) like this example:</p>
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<p> </p>
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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>Example: Soccer Game</h2>
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<p>You are off to soccer, and love being 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 the tree diagram. 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. This is done by multiplying each probability along the "branches" of the tree. </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 and include 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>Now we add the column:</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, it all adds up.</p>
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<p> </p>
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<p>You can see more uses of tree diagrams on <a href="probability-events-conditional.html">Conditional Probability</a>.</p>
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<h2>Conclusion</h2>
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<p>So there you go, when in doubt draw a tree diagram, multiply along the branches and add the columns. Make sure all probabilities add to 1 and you are good to go.</p>
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<p> </p>
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