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<title>Probability</title>
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<h1 align="center">Probability</h1>
<p class="center"><i>How <b>likely</b> something is to happen.</i></p>
<p>Many events can't be predicted with total certainty. The best we can say is how <b>likely</b> they are to happen, using the idea of probability.</p>
<h3>Tossing a Coin</h3>
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/head-tails-dollar.jpg" width="102" height="96" alt="head tails coin" /></p> <p>When a coin is tossed, there are two possible outcomes: </p>
<ul>
<li>heads (H) or </li>
<li>tails (T)</li>
</ul>
<p>We say that the probability of the coin landing <b>H</b> is <span class="large">&frac12;</span></p>
<p>And the probability of the coin landing <b>T</b> is <span class="large">&frac12;</span></p>
<p>&nbsp;</p>
<h3>Throwing Dice </h3>
<p style="float:left; margin: 0 10px 5px 0;"><img src="../geometry/images/pair-dice2.svg" alt="pair of dice" /></p> <p>When a single <a href="../geometry/fair-dice.html">die</a> is thrown, there are six possible outcomes: <b>1, 2, 3, 4, 5, 6</b>. </p>
<p>The probability of any one of them is <span class="intbl"><em>1</em><strong>6</strong>
</span></p>
<h2><span lang="EN-US" xml:lang="EN-US"> </span>Probability </h2>
<p>In general: </p>
<div class="def">
<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>
</div>
<p>&nbsp;</p>
<div class="example">
<h3>Example: the chances of rolling a &quot;4&quot; with a die</h3>
<p><b>Number of ways it can happen: 1</b> (there is only 1 face with a &quot;4&quot; on it)</p>
<p><b>Total number of outcomes: 6</b> (there are 6 faces altogether)</p>
<p class="center larger">So the probability = <span class="intbl">
<em>1</em>
<strong>6</strong>
</span></p>
</div>
<div class="example">
<h3>Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?</h3>
<p><b>Number of ways it can happen: 4</b> (there are 4 blues)</p>
<p><b>Total number of outcomes: 5</b> (there are 5 marbles in total)</p>
<p class="center larger">So the probability = <span class="intbl">
<em>4</em>
<strong>5</strong>
</span> = 0.8</p>
</div>
<h2>Probability Line</h2>
<p>We can show probability on a <a href="../probability_line.html">Probability Line</a>:</p>
<p class="center"><img src="images/probability-line.svg" alt="probability line" style="max-width:100%" /></p>
<p class="center large">Probability is always between 0 and 1</p>
<h2>Probability is Just a Guide</h2>
<p>Probability does not tell us exactly what will happen, it is just a guide</p>
<div class="example">
<h3>Example: toss a coin 100 times, how many Heads will come up?</h3>
<p>Probability says that heads have a <span class="large">&frac12;</span> chance, so we can <b>expect 50 Heads</b>.</p>
<p>But <b></b>when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.</p>
</div>
<p>Learn more at <a href="index.html#probability">Probability Index</a>.</p>
<h2>Words</h2>
<p>Some words have special meaning in Probability: </p>
</p> <div class="words">
<p><b>Experiment: </b>a repeatable procedure with a set of possible results.</p>
</div><div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/single-die.svg" alt="single die" /></p>
<h3>Example: Throwing dice</h3>
<p>We can throw the dice again and again, so it is repeatable.</p>
<p>The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}</p>
</div>
<p>&nbsp;
<div class="words">
<p><b>Outcome:</b> A possible result of an experiment.</p>
</div>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/single-die.svg" alt="single die" /></p> <h3>Example: Getting a &quot;6&quot;</h3>
</div>
<p>&nbsp;</p>
<div class="words">
<p><b>Sample Space:</b> all the possible outcomes of an experiment.</p>
</div>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/cards-hearts.svg" alt="hearts" /></p> <h3>Example: choosing a card from a deck</h3>
<p>There are 52 cards in a deck (not including Jokers)</p>
<p>So the<b> Sample Space is all 52 possible cards</b>: {Ace of Hearts, 2 of Hearts, etc... }</p>
</div>
<p>&nbsp;</p>
<p>The Sample Space is made up of Sample Points:</p>
<div class="words">
<p><b>Sample Point:</b> just one of the possible outcomes</p>
</div>
<div class="example">
<h3>Example: Deck of Cards</h3>
<ul>
<li>the 5 of Clubs is a sample point</li>
<li>the King of Hearts is a sample point</li>
</ul>
<p>&quot;King&quot; is not a sample point. There are 4 Kings, so that is 4 <i>different</i> sample points.</p>
</div>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/single-die.svg" alt="single die" /></p>
<h3>Example: Throwing dice</h3>
<p>There are 6 different sample points in the sample space.</p>
<p class="center"><img src="images/probability-sample-space.svg" alt="probability sample space" /></p>
</div>
<p>&nbsp;</p>
<div class="words">
<p><b>Event:</b> one <b>or more</b> outcomes of an experiment</p>
</div>
<div class="example">
<h3>Example Events:</h3>
<p>An event can be just one outcome:</p>
<ul>
<li>Getting a Tail when tossing a coin</li>
<li>Rolling a &quot;5&quot;</li>
</ul>
<p>An event can include more than one outcome:</p>
<ul>
<li>Choosing a &quot;King&quot; from a deck of cards (any of the 4 Kings)</li>
<li>Rolling an &quot;even number&quot; (2, 4 or 6)</li>
</ul>
</div>
<p>&nbsp;</p>
<p>Hey, let's use those words, so you get used to them:</p>
<div class="example">
<p style="float:right; margin: 10px;"><img src="../geometry/images/pair-dice2.svg" alt="pair of dice" /></p>
<h3>Example: Alex wants to see how many times a &quot;double&quot; comes up when throwing 2 dice.</h3>
<p>The <b>Sample Space</b> is all possible <b>Outcomes</b> (36 Sample Points): </p>
<p class="center">{1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6} </p>
<p>&nbsp;</p>
<p>The <b>Event</b> Alex is looking for is a &quot;double&quot;, where both dice have the same number. It is made up of these <b>6 Sample Points</b>:</p>
<p class="center"> {1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}</p>
<p>&nbsp;</p>
<p>These are Alex's Results:</p>
<div class="simple">
<table border="0" align="center">
<tr>
<th width="100" align="center">Experiment</th>
<th width="100" align="center">Is it a Double?</th>
</tr>
<tr>
<td align="center">{3,4}</td>
<td align="center">No</td>
</tr>
<tr>
<td align="center">{5,1}</td>
<td align="center">No</td>
</tr>
<tr>
<td align="center">{2,2}</td>
<td align="center"><b>Yes</b></td>
</tr>
<tr>
<td align="center">{6,3}</td>
<td align="center">No</td>
</tr>
<tr>
<td align="center">...</td>
<td align="center">...</td>
</tr>
</table>
</div>
<p>&nbsp;</p>
<p>After 100 <b>Experiments</b>, Alex has 19 &quot;double&quot; <b>Events</b> ... is that close to what you would expect?</p>
</div>
<p>&nbsp;</p>
<div class="questions">
<script type="text/javascript">getQ(700, 701, 702, 1475, 1476, 1477, 2175, 2176, 2177, 2178);</script>&nbsp;
</div>
<div class="activity"><a href="../activity/dice-experiment-1.html">An Experiment with a Die</a><br />
<a href="../activity/dice-experiment-2.html">An Experiment with Dice</a></div>
<div class="related"> <a href="index.html#probability">Probability Index</a> <a href="probability-events-types.html">Probability: Types of Events</a> <a href="../probability_line.html">Probability Line</a></div>
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