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<div id="main-column" class="span9"> <article id="probability-introduction-sn" style="margin-top: 16px;">
<h1 class="section-title">Probability introduction (SN)</h1>
<ul class="breadcrumb"><li><a title="Home" href="../../../index.html"><i class="fa fa-home"></i></a><span class="divider">/</span></li><li><span class="gray">4. Statistics & Probability</span><span class="divider">/</span></li><li><span class="active">Probability introduction (SN)</span></li></ul>
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<div class="yellowBg"><p><em>… It is remarkable that this science, which originated in the consideration of games of chance, should become the most important object of human knowledge … The most important questions in life are, for the most part, really only problems of probability. </em><br>Pierre Simon de Laplace (1749-1827)</p></div><h3><span style="color:#B22222;">∼ Student Notes ∼</span></h3><h3><span style="color:#B22222;">Section 4E.</span> Probability introduction</h3><p style="margin-left: 40px;"><u><img alt="" height="85" src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/prob_img11.jpg" style="float: left;" width="150"></u><strong>Sub-sections on this page:</strong><br><a class="scroll-to" data-target="origins"><u>4E.1</u> Origins</a><br><a class="scroll-to" data-target="terminology"><u>4E.2</u> Terminology</a><br><a class="scroll-to" data-target="flipping3"><u>4E.3</u> Flipping three coins</a><br><a class="scroll-to" data-target="theoret"><u>4E.4</u> Theoretical probability vs experimental probability</a></p><p style="margin-left: 40px;"></p><h4><strong><img alt="" src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/prob_img0a.jpg" style="float: right; width: 265px; height: 149px; border-width: 1px; border-style: solid;"><a class="anchor" id="origins" name="origins"> </a>4E.1 Origins</strong></h4><p>Games of chance, such as those that involve the rolling of dice, have been around for over 5000 years. For nearly as long as that, people have been trying to determine (or at least estimate) their chances (i.e. the probability) of winning such games. It’s interesting to note that the mathematical study of probability owes its origin almost entirely to efforts to answer questions about gambling. Although there were earlier attempts to study dice probabilities – including Girolamo Cardano (1501-1576) and Galileo Galilei (1564-1642) – it’s commonly accepted that modern concepts of probability originated in 1654 when two of the greatest mathematicians of the time, Blaise Pascal (1623-1662) and Pierre de Fermat (1607-1665), corresponded with each other when trying to solve some gambling problems that involved the rolling of dice. Some decades later, Pierre-Simon Laplace (1749-1827), made a significant contribution to the development of the mathematical theory of probability with an emphasis on scientific applications in his 1812 book <em>Théorie analytique des probabilités</em> (<em>Analytic theory of probability</em>). In the 19<sup>th</sup> century, James Clerk Maxwell (1831-1879) and Ludwig Boltzmann (1844-1906) introduced the ideas of probability and random events into the laws of physics with the kinetic theory of gases. Important aspects of the laws of quantum mechanics, developed in the first half of the 20<sup>th</sup> century, relied on the mathematics of probability.</p><div class="polaroid-left"><img src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/head_tail_img1.jpg" style="margin:8px 0"><div class="caption">A British 50 pence coin is a seven-side coin that <br>is formed as an equilateral-curve heptagon, or<br>Reuleaux polygon, a curve of constant width.</div></div><h4><a class="anchor" id="terminology" name="terminology"> </a>4E.2 Terminology</h4><p>Let’s consider flipping a single coin. Unless stated otherwise we assume a coin (or a die) is <em>fair </em>which means that all possible results, or <strong>outcomes</strong> (landing ‘heads’ or ‘tails’ in the case of a coin) are <strong>equally likely</strong>. Flipping a coin is an example of a <strong>random experiment </strong>because the possible outcomes (head or tail) are equally likely resulting only from chance. Each repetition of an experiment is a <strong>trial</strong>. The set of all possible outcomes for an experiment is called the <strong>sample space</strong>. An outcome, or set of outcomes, is considered an <strong>event</strong> and commonly labelled with a capital letter.<em> </em>For example, <em>H</em> can represent obtaining a 'head' when flipping a coin. The <strong>probability </strong>of an event occurring is a ratio that compares how many ways it can occur to how many way all the outcomes in the sample space can occur. Since the probability is for an event associated with a <em>random </em>experiment, we consider the probability to be <strong>theoretical </strong>since it expresses what we <u>expect</u> to happen. If we flip a fair coin 1000 times we expect that 500 of these will show 'head' but understand that this is not certain. However, it makes sense that as we flip the coin more and more the ratio of 'head' results to the total # of flips will get closer and closer to <span class="math-tex">\(\frac{1}{2}\)</span>. This ratio that uses observed data from multiple trials of a random experiment is not a theoretical probability, but an <strong>experimental probability</strong>.</p><div class="blueBg"><p><strong>Theoretical probability of an event</strong><br>If all outcomes in the sample space of a random experiment are equally likely, then the theoretical probablity of event <em>A</em> is given by <img alt="" src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/prob_formula2_img.jpg" style="width: 410px; height: 50px;"></p><p>where <span class="math-tex">\({\textrm{0}} \le {\textrm{P}}\left( A \right) \le 1\)</span>. If <span class="math-tex">\({\textrm{P}}\left( A \right) = 1\)</span>, then event <em>A</em> is certain to occur; and if <span class="math-tex">\({\textrm{P}}\left( A \right) = 0\)</span>, then it is impossible for event <em>A</em> to occur.</p></div><h4><img alt="" src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/s_space1_img.jpg" style="float: right; width: 50px; height: 178px;"><a class="anchor" id="flipping3" name="flipping3"> </a>4E.3 Flipping three coins</h4><p>Before we look at a couple of the dice games that lead Pascal and Fermat to develop a mathematical method for determining probabilities, let’s consider a simpler game that involves simultaneously flipping three coins - and use the above definition to compute the probability of two different events. <strong>Event <em>A</em> is all three coins show 'heads'</strong>; and <strong>event <em>B</em> is <em>at least</em> two of the coins show 'heads'</strong>. Determining the <strong>sample space</strong> for an experiment is essential for finding the probability of an event that is a subset of the sample space. The sample space is the same for events <em>A</em> and <em>B</em>. The eight possible outcomes when flipping three coins are listed in the figure at right. Note that event <em>A</em> is a single outcome (HHH) and event <em>B</em> consists of more than one outcome. It is clear that <span class="math-tex">\({\textrm{P}}\left( A \right) = \frac{1}{8}\)</span> because there is only one way for three heads to occur and there are eight equally likely outcomes in the sample space. What about <span class="math-tex">\({\textrm{P}}\left( B \right)\)</span>? Remember, event B is getting <u>at least</u> two heads when three coins are flipped. Well, one outcome (HHH) has three heads and three other outcomes (HHT, HTH, THH) have two heads. These can be referred to as the <strong>favourable outcomes</strong> with regard to getting at least two heads. Thus, there are four ways for <strong>at least</strong> two heads to occur which means that <span class="math-tex">\({\textrm{P}}\left( B \right) = \frac{4}{8} = \frac{1}{2}\)</span>.</p><p>Explore what occurs when flipping three coins simultaneously with the Geogebra applet below. You can input how many flips of the three coins to be carried out. The # of flips is the # of trials for this <strong>random experiment</strong> of flipping three coins. Results are displayed in a histogram on the right side. How does the <strong>theoretical probability</strong> for events <em>A</em> and <em>B</em> compare to the <strong>experimental probability</strong> for multiple trials of flipping three coins with the applet?</p><p><iframe height="484px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/kcvky3fv/width/673/height/484/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="flipping three coins" width="673px"></iframe></p><h4><a class="anchor" id="theoret" name="theoret"> </a>4E.4 Theoretical probability versus experimental probability</h4><p>As discussed above in connection to the definition that was presented for the theoretical probability of an event, the calculation of an event's theoretical probability is based on the fact that all possible outcomes in the experiment's sample space are equally likely. If the <span class="math-tex">\(n\)</span> possible outcomes for an experiment are all equally likely then the probability of one of the possible outcomes occurring is <span class="math-tex">\(\frac{1}{n}\)</span>. If an event consists of more than one outcome (as happened with event B above: getting at least 2 heads when flipping 3 coins), then the probability is the number of those 'favourable' outcomes (i.e. the # of outcomes that produce the result you're looking for) divided by the # of all possible outcomes in the sample space (sometimes referred to as the 'size' of the sample space). Hence, the 'formula' for the probability of an event A presented in the <a href="/media/ib/mathanalysis/analysis/basics/aa-formula-bklet-v1.2.pdf" target="_blank">Formula Booklet</a> for this course is for the theoretical probability of event A.</p><div class="blueBg"><p><strong>Analysis & Approaches Formula Booklet</strong> (theoretical probability)</p><p style="margin-left: 40px;">Probability of an event <span class="math-tex">\(A\)</span> (with sample space <span class="math-tex">\(U\)</span>): <span class="math-tex">\({\textrm{P}}\left( A \right) = \frac{{n\left( A \right)}}{{n\left( U \right)}}\)</span></p><p>where <span class="math-tex">\(n\left( A \right)\)</span> is the # of outcomes that result in event A occurring ('favoruable' outcomes), and <span class="math-tex">\(n\left( U \right)\)</span> is the # of possible outcomes (size of the sample space)</p></div><p>However, there is another way to try and calculate the probability of an event; especially if the outcomes in the sample space are not all equally likely and/or if it is difficult (or impossible) to determine all the outcomes in the sample space. In this situation, we coud repeat the random experiment many, many times (i.e. for a large # of trials) and simply count how many times (frequency) we obtain a favourable outcome. In this approach, we are estimating the probability of an event by finding the <strong>relative frequency</strong> - or <strong>experimental probability</strong> - of the event in repeated trials of the indicated experiment. As the # of repeated trials increases, the <em>limit </em>of the relative frequency will become a better and better estimate of the event's theoretical probability. This <strong><em>experimental probability approach</em></strong> requires the capability of repeating a random experiment for a reasonably large # of trials. This is not always easy to do. This is one reason why calculating an experimental probability will <u><strong>not</strong></u> be the method that you will use to answer any probability question on a quiz, test or exam for this course. However, you may decide to use this approach in an Exploration (IA) that explores a probability topic.</p><div class="yellowBg"><h5><img alt="" src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/buffon_img2.jpg" style="float: right; width: 150px; height: 193px;">Repeatedly tossing a coin and experimental probability</h5><p>In the 18th century the French mathematician and naturalist Georges-Louis Leclerc, count de Buffon, (known for the famous <a href="https://en.wikipedia.org/wiki/Buffon%27s_needle_problem" target="_blank">Buffon's needle problem</a>) tossed a coin 4040 times resulting in 2048 heads. The relative frequency of approximately 0.50693. In the early 20th century, the English mathematician Karl Pearson tossed a coin 24,000 times and obtained 12,012 heads. The relative frequency of Pearson’s experiment was 0.5005, even closer to the theoretical probability of 0.5 for a fair coin.</p><p>Another way of conducting the random experiment of repeatedly tossing a coin to determine the experimental probability (relative frequency) of obtaining heads is to write a computer/calculator program that makes use of a random number generator to "toss a coin" (e.g. randomly generate a 0 or a 1 where a 0 can represent 'tails' and a 1 can represent 'heads'). Such a program was written for the TI-Nspire that modelled flipping a coin 7350 times. The calculator's graph window can be divided into 105 columns and 70 rows creating 7350 small squares (105 x 70 = 7350). Each square represents the result of one of the coin tosses: tails (0) makes a square blank (white), and heads (1) fills in a square (blue). The image below shows one result of running this program on the TI-Nspire; and that the relative frequency of heads was <span class="math-tex">\(\frac{{3655}}{{7350}} \approx 0.497279\)</span></p><p><img alt="" src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/nspire_coin_img1.jpg" style="float: left; width: 400px; height: 300px;"></p><p><img alt="" src="../../../ib/mathanalysis/analysis/stats---probability/probability-student/nspire_coin_prgm_results.jpg" style="width: 250px; height: 96px;"></p><p>What if repeatedly tossing a coin resulted in a relative frequency of heads (or tails) that did not approach the theoretical probability of 0.5? In that case, we would have to conclude that the empirical evidence is showing that the coin is probably not a <em><strong>fair</strong></em> coin; there is a greater (or lesser) chance of getting heads than tails.</p></div><script>document.querySelectorAll('.tib-teacher-only').forEach(e => e.remove());</script>
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