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300 lines
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300 lines
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<h1 class="center">Probability: Independent Events</h1>
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<p><img src="images/random-life.jpg" style="max-width:100%" alt="random life"></p>
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<p>Life is full of random events!</p>
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<p>You need to get a "feel" for them to be a smart and successful person.</p>
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<p>The toss of a coin, throwing dice and lottery draws are all examples of random events. </p>
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<p>There can be:</p>
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<div class="dotpoint"><b>Dependent Events</b> where what happens <b>depends on</b> what happened before, such as taking cards from a deck makes less cards each time (learn more at <a href="probability-events-conditional.html">Conditional Probability</a>), or</div>
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<div class="dotpoint"><b>Independent Events</b> which we learn about here.</div>
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<h2>Independent Events</h2>
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<p style="float:left; margin: 0 30px 5px 0;"><img src="images/probability-coin-toss.svg" alt="probability coin toss"></p>
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<p>Independent Events are <b>not affected</b> by previous events. </p>
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<p class="center"><b>This is an important idea!</b></p>
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<p>A coin does not "know" it came up heads before. </p>
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<p>And each toss of a coin is a perfect isolated thing.</p>
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<div class="example">
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<h3>Example: You toss a coin and it comes up "Heads" three times ... what is the chance that <b>the next toss</b> will also be a "Head"?</h3>
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<p><b>The chance is simply <span class="large">½</span> (or 0.5) just like ANY toss of the coin.</b></p>
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<p>What it did in the past will not affect the current toss!</p>
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</div>
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<p>Some people think "it is overdue for a Tail", but <i>really truly</i> the next toss of the coin is totally independent of any previous tosses.</p>
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<p class="center80">Saying "a Tail is due", or "just one more go, my luck is due to change" is called <b>The Gambler's Fallacy</b> </p>
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<p>Of course your luck <b>may</b> change, because each toss of the coin has an equal chance.</p>
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<h2>Probability of Independent Events</h2>
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<p class="words">"Probability" (or "Chance") is <b>how likely</b> something is to happen.</p>
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<p>So how do we calculate probability? </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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<p style="float:left; margin: 0 30px 10px 0;"><img src="images/probability-coin-toss.svg" alt="probability coin toss">
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</p>
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<h3>Example: what is the probability of getting a "Head" when tossing a coin?</h3>
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<p><b>Number of ways it can happen: 1</b> (Head)</p>
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<p><b>Total number of outcomes: 2</b> (Head and Tail)</p>
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<p class="center larger">So the probability = <span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span> = 0.5</p>
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</div>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/dice-cube.jpg" alt="dice cube" height="150" width="154"></p>
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<h3>Example: what is the probability of getting a "4" or "6" when rolling a die?</h3>
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<p><b>Number of ways it can happen: 2</b> ("4" and "6")</p>
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<p><b>Total number of outcomes: 6</b> ("1", "2", "3", "4", "5" and "6")</p>
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<p class="center larger">So the probability = <span class="intbl">
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<em>2</em>
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<strong>6</strong>
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</span> = <span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> = 0.333... </p>
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</div>
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<h2>Ways of Showing Probability</h2>
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<p><a href="../probability_line.html">Probability</a> goes from <b>0</b> (imposssible) to <b>1</b> (certain):</p>
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<p class="center"><img src="images/probability-line.svg" alt="probability line" style="max-width:100%"></p>
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<p class="larger">It is often shown as a <b>decimal</b> or <b>fraction</b>.</p>
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<div class="example">
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<h3>Example: the probability of getting a "Head" when tossing a coin:<b></b></h3>
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<ul>
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<li>As a decimal: <b>0.5</b></li>
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<li>As a fraction: <b>1/2</b></li>
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<li>As a percentage: <b>50%</b></li>
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<li>Or sometimes like this: <b>1-in-2</b></li>
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</ul>
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</div>
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<h2>Two or More Events</h2>
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<p>We can calculate the chances of two or more <b>independent</b> events by <b>multiplying</b> the chances.</p>
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<div class="example">
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<h3>Example: Probability of 3 Heads in a Row</h3>
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<p>For each toss of a coin a Head has a probability of 0.5:</p>
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<p class="center"><img src="images/probability-coin-hhh.svg" alt="probability coin hhh = 0.5x0.5x0.5 = 0.125"></p>
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<p>And so the chance of getting <span class="larger">3 Heads in a row is <b>0.125</b></span></p>
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</div>
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<p>So each toss of a coin has a <span class="large">½</span> chance of being Heads, but<b> lots of Heads in a row</b> is unlikely.</p>
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<div class="example">
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<h3>Example: Why is it unlikely to get, say, 7 heads in a row, when <i>each</i> toss of a coin has a <span class="large">½</span> chance of being Heads?</h3>
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<p>Because we are asking two different questions:</p>
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<p>Question 1: What is the probability of 7 heads in a row? </p>
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<div class="so">Answer: <span class="intbl"><em>1</em><strong>2</strong></span>×<span class="intbl"><em>1</em><strong>2</strong></span>×<span class="intbl"><em>1</em><strong>2</strong></span>×<span class="intbl"><em>1</em><strong>2</strong></span>×<span class="intbl"><em>1</em><strong>2</strong></span>×<span class="intbl"><em>1</em><strong>2</strong></span>×<span class="intbl"><em>1</em><strong>2</strong></span> = <b>0.0078125</b> (less than 1%)</div>
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<!-- Answer: 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 0.0078125 (less than 1%). -->
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<p>Question 2: When <b>we have just got 6 heads</b> in a row, what is the probability that <b>the next toss</b> is also a head? </p>
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<div class="so">Answer: <span class="large">½</span>, as the <b>previous</b> tosses don't affect the next toss</div>
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</div>
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<p>You can have a play with the <a href="quincunx.html">Quincunx</a> to see how lots of independent effects can still have a pattern.</p>
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<h2>Notation</h2>
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<p>We use "P" to mean "Probability Of",</p>
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<p>So, for Independent Events:</p>
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<p class="larger center">P(A and B) = P(A) × P(B)</p>
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<p class="center">Probability of A and B equals the probability of A times the probability of B</p>
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<div class="example">
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<h3>Example: your boss (to be fair) randomly assigns everyone an extra 2 hours work on weekend evenings between 4 and midnight.</h3>
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<p>What are the chances you get Saturday between 4 and 6?</p>
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<p class="center"><img src="images/probability-independent-times.svg" alt="probability independent times"></p>
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<p><b>Day:</b> there are two days on the weekend, so <b>P(Saturday) = 0.5</b></p>
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<p><b>Time:</b> you want the <b>2 hours</b> of "4 to 6", out of the <b>8 hours</b> of 4 to midnight): </p>
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<p class="center"><b>P("4 to 6") = 2/8 = 0.25</b></p>
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<p>And:</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="center"><b>P(Saturday and "4 to 6")</b></span></td>
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<td nowrap="nowrap"><span class="center">= P(Saturday) × P("4 to 6")</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="center">= 0.5 × 0.25</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="center">= <b>0.125</b></span></td>
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</tr>
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</tbody></table>
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<p class="larger center">Or a 12.5% Chance</p>
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<p>(Note: we could ALSO have worked out that you wanted 2 hours out of a total possible 16 hours, which is 2/16 = 0.125. Both methods work here.)</p>
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</div>
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<h2>Another Example</h2>
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<div class="example">
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<h3>Example: the chance of a flight being delayed is 0.2 (=20%), what are the chances of no delays on a round trip</h3>
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<p>The chance of a flight <b>not</b> having a delay is 1 − 0.2 = <b>0.8</b>, so these are all the possible outcomes:</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>0.8 × 0.8 =</td>
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<td> </td>
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<td><b>0.64</b> chance of <b>no delays</b></td>
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</tr>
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<tr>
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<td>0.2 × 0.8 = </td>
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<td> </td>
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<td><b>0.16</b> chance of 1st flight delayed</td>
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</tr>
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<tr>
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<td>0.8 × 0.2 =</td>
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<td> </td>
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<td><b>0.16</b> chance of return flight delayed</td>
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</tr>
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<tr>
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<td>0.2 × 0.2 =</td>
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<td> </td>
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<td><b>0.04</b> chance of both flights delayed</td>
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</tr>
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</tbody></table>
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<p>When we add all the possibilities we get: </p>
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<p class="center">0.64 + 0.16 + 0.16 + 0.04 = <b>1.0</b></p>
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<p>They all add to 1.0, which is a good way of checking our calculations.</p>
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<p class="larger center">Result: <b>0.64</b>, or a 64% chance of no delays</p>
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</div>
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<h2>One More Example</h2>
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<p>Imagine there are two groups: </p>
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<ul>
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</ul>
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<ul>
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<li>A member of each group gets randomly chosen for the winners circle, </li>
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<li><b>then</b> one of those gets randomly chosen to get the big money prize:</li>
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</ul>
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<p class="center"><img src="images/probability-winners.svg" alt="probability winners" style="max-width:100%"></p>
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<p>What is your chance of winnning the big prize?</p>
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<ul>
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<li>there is a <b>1/5 chance</b> of going to the winners circle </li>
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<li>and a <b>1/2 chance</b> of winning the big prize</li>
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</ul>
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<p>So you have a 1/5 chance followed by a 1/2 chance ... which makes a 1/10 chance overall:</p>
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<p class="center larger"><span class="intbl"><em>1</em><strong>5</strong></span> × <span class="intbl"><em>1</em><strong>2</strong></span> = <span class="intbl"><em>1</em><strong>5 × 2</strong></span> = <span class="intbl"><em>1</em><strong>10</strong></span></p>
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<p>Or we can calculate using decimals (1/5 is 0.2, and 1/2 is 0.5):</p>
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<p class="center larger">0.2 x 0.5 = <b>0.1</b></p>
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<p>So your chance of winning the big money is <b>0.1</b> (which is the same as 1/10).</p>
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<h2>Coincidence!</h2>
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<p>Many "Coincidences" are, in fact, likely.</p>
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<div class="example">
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<h3>Example: you are in a room with 30 people, and find that Zach and Anna celebrate their birthday on the same day. </h3>
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<p>Do you say:</p>
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<ul>
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<li>"Wow, how strange !", or </li>
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<li>"That seems reasonable, with so many people here"</li>
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</ul>
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<p>In fact there is a <b>70% chance</b> that would happen ... so it is <b>likely</b>.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/probability-many-many.svg" alt="probability many to many"></p>
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<p class="larger">Why is the chance so high?</p>
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<p>Because you are comparing everyone to everyone else (not just one to many).</p>
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<p>And with 30 people that is <b>435 comparisons</b></p>
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<p> </p>
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<p>(Read <a href="probability-shared-birthday.html">Shared Birthdays</a> to find out more.)</p>
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</div><p> </p>
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<div class="example">
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<h3>Example: Snap! </h3>
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<p>Did you ever say something at exactly <b>the same time as someone else</b>?</p>
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<p>Wow, how amazing!</p>
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<p><b>But</b> you were probably sharing an experience (movie, journey, whatever) and so your thoughts were similar. </p>
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<p>And there are only so many ways of saying something ...</p>
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<p class="center">... so it is like the card game "Snap!" (also called Slaps or Slapjack) ...</p>
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<p class="center">... if you speak enough words together, they will eventually match up.</p>
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<p>So, maybe not so amazing, just simple chance at work.</p>
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</div>
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<p>Can you think of other cases where a "coincidence" was simply a likely thing?</p>
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<h2>Conclusion</h2>
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<ul>
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<div class="larger">
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<li>Probability is: (Number of ways it can happen) / (Total number of outcomes)</li>
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<li>Dependent Events (such as removing marbles from a bag) are affected by previous events</li>
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<li>Independent events (such as a coin toss) are <b>not</b> affected by previous events</li>
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<li>We can calculate the probability of two or more <b>Independent</b> events by <b>multiplying</b></li>
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<li>Not all coincidences are really unlikely (when you think about them).</li>
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<p> </p>
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