{"id":297,"date":"2017-08-24T07:08:04","date_gmt":"2017-08-24T07:08:04","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=297"},"modified":"2017-09-13T08:02:54","modified_gmt":"2017-09-13T08:02:54","slug":"independent-and-dependent-events","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/independent-and-dependent-events\/","title":{"rendered":"Independent and Dependent Events"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/utilizing-combinations-and-permutations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/probability-statistics-data-analysis\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/beginning-statistics\">Next\u00a0Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Independent and Dependent Events<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study the difference between dependent and independent events and how to use conditional\u00a0probability.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li><strong><em>Probability <\/em><\/strong>is\u00a0the likelihood that an event will occur.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p1_clip_image003.gif\" width=\"191\" height=\"18\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p1_clip_image006.gif\" width=\"188\" height=\"18\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p1_clip_image009.gif\" width=\"137\" height=\"18\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<section>\n<h3>What is the difference between independent and dependent\u00a0events?<\/h3>\n<p><span style=\"text-decoration: none;\">Sometimes the probability\u00a0of an event can change, given the number of previous trials or\u00a0information about other events. Two events are <\/span><abbr title=\" Two events are independent if the probability of the second event is not altered by the outcome of the first event.. \"><span style=\"text-decoration: none;\">independent<\/span><\/abbr><span style=\"text-decoration: none;\"> if the probability of the second event is not affected by the\u00a0outcome of the first event. If, instead, the outcome of the first\u00a0event <\/span><em><span style=\"text-decoration: none;\">does<\/span><\/em><span style=\"text-decoration: none;\"> affect the probability of the second event, these events are <\/span><abbr title=\" Events are dependent if the probability of the second event is altered by the outcome of the first event. \"><span style=\"text-decoration: none;\">dependent<\/span><\/abbr><span style=\"text-decoration: none;\">.<\/span><\/p>\n<p style=\"text-decoration: none;\">Examples of independent events:<\/p>\n<ul>\n<li>flipping\u00a0a coin and rolling a die<\/li>\n<li>flipping a coin or rolling a die twice<\/li>\n<li>choosing a card from a deck, replacing it, shuffling, and choosing again<\/li>\n<li>choosing a particular pair\u00a0of shoes and the weather in Sweden<\/li>\n<\/ul>\n<p>Examples of dependent events:<\/p>\n<ul>\n<li>choosing\u00a0a card from a deck and then choosing another without putting the\u00a0first card back in the deck<\/li>\n<li>picking\u00a0two marbles from a bag that contains ten black marbles and ten\u00a0white marbles without putting either marble back in the bag<\/li>\n<li>what I choose to eat for\u00a0lunch and whether I am hungry by three o&#8217;clock<\/li>\n<\/ul>\n<h3>Independent and Dependent Events<\/h3>\n<p>Imagine a bag containing 5 red marbles, 3 white marbles, and 2\u00a0blue marbles. We pick two marbles from the bag, first one and then\u00a0the other, trying to get two of the same color. This scenario can\u00a0produce either independent or dependent events.<\/p>\n<p><span style=\"text-decoration: none;\">If we replace the marble we\u00a0chose first and mix the marbles before making our second choice,\u00a0then the second choice will be independent from the first.\u00a0However, if we <\/span><em><span style=\"text-decoration: none;\">do\u00a0not <\/span><\/em><span style=\"text-decoration: none;\">replace the\u00a0first marble before choosing the second, this alters the\u00a0probability of the second pick by changing the number of possible\u00a0outcomes: there is one fewer possibility. <\/span><\/p>\n<p>When computing the probability of two events, it is important\u00a0to note whether the events are independent or dependent. The\u00a0event&#8217;s status can change depending on the situation.<\/p>\n<p>Consider the bag of marbles above. What is the probability of\u00a0choosing two white marbles, one after the other, if:<\/p>\n<p>(a) we replace the first marble before choosing a second\u00a0marble?<br \/>\n(b) we do not replace the first marble before choosing a second\u00a0marble?<\/p>\n<p><span style=\"text-decoration: none;\">Let <\/span><em><span style=\"text-decoration: none;\">A <\/span><\/em><span style=\"text-decoration: none;\">= &#8220;the first\u00a0marble is white&#8221; and <\/span><em><span style=\"text-decoration: none;\">B <\/span><\/em><span style=\"text-decoration: none;\">= &#8220;the second\u00a0marble is white.&#8221; <\/span><\/p>\n<p style=\"text-decoration: none;\">Case (a): in which the first\u00a0marble is replaced\u2014is an example of two independent events. Notice that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p3_clip_image003.gif\" width=\"119\" height=\"34\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0because the two events occur under the exact same circumstances.<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p3_clip_image006.gif\" width=\"279\" height=\"39\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Case (b)\u2014in which the first marble is not replaced\u2014is\u00a0an example of a dependent event. If we pick a marble and do not\u00a0replace it, the sample space has changed. We still have <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p3_clip_image009.gif\" width=\"67\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0However, for the second drawing, the sample space consists of only\u00a09 marbles and<span style=\"text-decoration: none;\">, assuming that <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> was successful, there are only 2 white marbles left in the bag.\u00a0Thus, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p3_clip_image012.gif\" width=\"61\" height=\"34\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p3_clip_image015.gif\" width=\"298\" height=\"34\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Thus we know that case (a) is more probable than case (b). <\/span><\/p>\n<div class=\"callout\">\n<h4>Be Aware!<\/h4>\n<p>When calculating the probability of dependent events, we can simplify\u00a0matters by\u00a0assuming that the first event resulted in the desired outcome when computing the probability of the\u00a0second event.\u00a0Keep this in mind when calculating dependent probabilities.<\/p>\n<p class=\"notebox_text\" align=\"LEFT\"><span style=\"text-decoration: none;\">For example, when computing probabilities of various choices from the bag of marbles, we assumed that event <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> was successful (had a favorable outcome, i.e., that a white marble was drawn). Then 2 white marbles would remain in the bag, and the probability of drawing another is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p3_clip_image017.gif\" width=\"61\" height=\"34\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/span><\/p>\n<\/div>\n<h3>Independent and Dependent Events<\/h3>\n<p>If we know that an event has\u00a0already occurred and we know its outcome, how does this alter the\u00a0probability of another event\u2019s outcome?<\/p>\n<p><span style=\"text-decoration: none;\">The answer to this question\u00a0depends on whether the two events are dependent or independent. In\u00a0general, <\/span><abbr title=\"measures the probability of an event B provided we know another event A has already occurred; it is calculated using the formula MathType. \"><span style=\"text-decoration: none;\">conditional\u00a0probability<\/span><\/abbr><span style=\"text-decoration: none;\"> measures the probability of an outcome <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\"> provided we know that another outcome <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> has already occurred. <\/span><\/p>\n<p style=\"text-decoration: none;\">The formula for conditional probability is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p4_clip_image003.gif\" width=\"49\" height=\"21\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p><span style=\"text-decoration: none;\">The idea of conditional\u00a0probability is closely related to the difference between the\u00a0probability of dependent events and the probability of independent\u00a0events. Recall that when computing the probability of two\u00a0dependent events <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> and <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0we can assume that <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> was successful to simplify computing the probability of B. Because\u00a0we make this assumption, we really compute is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p4_clip_image006.gif\" width=\"186\" height=\"21\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We can rearrange this formula to get the equation we will use for\u00a0conditional probability. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">If the two events are\u00a0independent then it does not matter whether <\/span><em><span style=\"text-decoration: none;\">A <\/span><\/em><span style=\"text-decoration: none;\">has occurred,\u00a0because <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> has no influence on the probability of <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0Thus <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p4_clip_image009.gif\" width=\"101\" height=\"21\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p><span style=\"text-decoration: none;\">If <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> and <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\"> are the outcomes of <\/span><em><span style=\"text-decoration: none;\">dependent<\/span><\/em><span style=\"text-decoration: none;\"> events, then <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p4_clip_image011.gif\" width=\"186\" height=\"21\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\nand thus <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p4_clip_image014.gif\" width=\"146\" height=\"41\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p class=\"notebox_text\" align=\"center\"><span style=\"text-decoration: none;\">If <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> and <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\"> are the outcomes of <\/span><em><span style=\"text-decoration: none;\">independent<\/span><\/em><span style=\"text-decoration: none;\"> events, then <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p4_clip_image016.gif\" width=\"101\" height=\"21\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\nand <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p4_clip_image019.gif\" width=\"171\" height=\"18\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<p><!-- VIDEO BROKEN?\n\n<strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/videoicon.jpg\" name=\"graphics10\" alt=\"video icon\" align=\"BOTTOM\" width=\"60\" height=\"38\" border=\"0\">Classroom\nDemonstration Video<\/strong>\n<span class=\"fineprint_ital\">In order to view this\nvideo, you must have RealPlayer installed on your computer. If you\ndo not, you may download it for free from <a href=\"http:\/\/www.real.com\/\" target=\"_blank\">this\nsite<\/a>. <\/span>\n\nTo watch this demonstration of probability with a\ndial-up connection click <a href=\"http:\/\/www.abcte.org\/bb-data\/mt01-lo.ram\">here<\/a>.\nTo watch with a high-speed connection click <a href=\"http:\/\/www.abcte.org\/bb-data\/mt01-hi.ram\">here<\/a>.\n\n--><\/p>\n<h3>What if there are more than two\u00a0events to consider?<\/h3>\n<p><span style=\"text-decoration: none;\">When we need to consider <\/span><em><span style=\"text-decoration: none;\">more<\/span><\/em><span style=\"text-decoration: none;\"> than two events, we can use induction based on what we already\u00a0know about two events. Suppose we have three events and three\u00a0favorable outcomes <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\">, <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0and <\/span><em><span style=\"text-decoration: none;\">C<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0We want to compute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p5_clip_image003.gif\" width=\"119\" height=\"18\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0To make sense of this scenario, we can rewrite it and formulate\u00a0the result using what we know about two events. Say <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p5_clip_image006.gif\" width=\"270\" height=\"21\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p><span style=\"text-decoration: none;\">If <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\">,<\/span><em><span style=\"text-decoration: none;\"> B<\/span><\/em><span style=\"text-decoration: none;\">, and <\/span><em><span style=\"text-decoration: none;\">C<\/span><\/em><span style=\"text-decoration: none;\"> are all independent, then <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p5_clip_image009.gif\" width=\"410\" height=\"21\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p class=\"notebox_text\" align=\"LEFT\"><span style=\"text-decoration: none;\">If <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\">, <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\">, and <\/span><em><span style=\"text-decoration: none;\">C<\/span><\/em><span style=\"text-decoration: none;\"> are all dependent, then <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p5_clip_image012.gif\" width=\"558\" height=\"23\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p><span style=\"text-decoration: none;\">Suppose we choose two cards\u00a0from a deck of 52, making our choices one at a time without\u00a0replacing cards. What is the probability of the first card being a\u00a0black face card <\/span><em><span style=\"text-decoration: none;\">and<\/span><\/em><span style=\"text-decoration: none;\"> the second card being a heart? <\/span><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p5_clip_image015.gif\" width=\"17\" height=\"34\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p5_clip_image018.gif\" width=\"15\" height=\"34\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p5_clip_image021.gif\" width=\"17\" height=\"34\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p5_clip_image024.gif\" width=\"17\" height=\"34\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct choice is A.\u00a0Let <\/span><em><span style=\"text-decoration: none;\">A <\/span><\/em><span style=\"text-decoration: none;\">=\u00a0&#8220;the first card drawn is a black face card&#8221; and <\/span><em><span style=\"text-decoration: none;\">B <\/span><\/em><span style=\"text-decoration: none;\">= &#8220;the\u00a0second card drawn is a heart.&#8221; Because we did not replace\u00a0the first card we chose, these two events are dependent, and thus <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p5_clip_image027.gif\" width=\"220\" height=\"34\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p><span style=\"text-decoration: none;\">The local bookstore often\u00a0gets new shipments of math books. In fact, the probability that\u00a0they will receive new math books on any given day is 76%. If a\u00a0shipment of new math books does arrive, you expect that there is\u00a0a 30% probability that the particular book you want to buy will\u00a0be in that shipment. What is the probability that there will be a\u00a0shipment of new math books today <\/span><em><span style=\"text-decoration: none;\">and<\/span><\/em><span style=\"text-decoration: none;\"> that the book you want to buy will be in that shipment? <\/span><\/p>\n<table border=\"0\" width=\"40%\" cellpadding=\"10\">\n<tbody>\n<tr>\n<td>\n<div align=\"center\">A<\/div>\n<\/td>\n<td>22.8%<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">B<\/div>\n<\/td>\n<td>25%<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">C<\/div>\n<\/td>\n<td>26.2%<\/td>\n<\/tr>\n<tr>\n<td>\n<div align=\"center\">D<\/div>\n<\/td>\n<td>28%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is A.\u00a0Let <em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> = &#8220;a shipment of new math books arrives today&#8221; and <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\"> = &#8220;the book you want to buy is in that shipment.&#8221; The\u00a0question asks us to determine the probability that new math\u00a0books arrive <\/span><em><span style=\"text-decoration: none;\">and<\/span><\/em><span style=\"text-decoration: none;\"> that the book you want to buy is among them, in other words, we\u00a0want <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image003.gif\" width=\"76\" height=\"18\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>. <\/span><\/p>\n<p style=\"text-decoration: none;\">We know that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image006.gif\" width=\"123\" height=\"18\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Given the events described above, we get <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image009.gif\" width=\"25\" height=\"19\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>=\u00a0&#8220;provided the bookstore receives new math books today, your\u00a0book will be among them.&#8221; The question tells us that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image012.gif\" width=\"138\" height=\"21\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Therefore, using our equation for conditional probability, we\u00a0get<\/p>\n<p style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image015.gif\" width=\"288\" height=\"115\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<div class=\"callout\">\n<h4>Be Aware!<\/h4>\n<p>When solving probability questions, it helps to keep track of the events and their\u00a0outcomes. To ensure that this information is clear, in the very beginning, always define the outcomes of the events\u00a0with which you are working . Write them down so that you can refer back to your notes if you get confused.<\/p>\n<\/div>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>There are 75 students in a freshman chemistry class at\u00a0university Q. Before the first exam, the students answered a\u00a0questionnaire about whether they had studied and the following\u00a0table was compiled from their responses and their grades.<\/p>\n<table width=\"85%\">\n<tbody>\n<tr>\n<th colspan=\"3\">Chemistry Class<\/th>\n<\/tr>\n<tr>\n<td width=\"39%\"><\/td>\n<td width=\"26%\">\n<div align=\"center\">Students who studied<\/div>\n<\/td>\n<td width=\"35%\">\n<div align=\"center\">Students who didn\u2019t study<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td># of students who passed<\/td>\n<td>\n<div align=\"center\">55<\/div>\n<\/td>\n<td>\n<div align=\"center\">6<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td># of students who failed<\/td>\n<td>\n<div align=\"center\">2<\/div>\n<\/td>\n<td>\n<div align=\"center\">12<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What is the probability that a random student who did not study passed the exam?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image018.gif\" width=\"16\" height=\"34\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image021.gif\" width=\"9\" height=\"34\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image024.gif\" width=\"9\" height=\"34\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image027.gif\" width=\"17\" height=\"34\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct choice is B.\u00a0Let <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> = &#8220;this student did not study&#8221; and <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\"> = &#8220;this student passed the test.&#8221; We want to find <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image030.gif\" width=\"49\" height=\"21\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and to find this quantity we will use <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image033.gif\" width=\"146\" height=\"41\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>. <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image036.gif\" width=\"430\" height=\"34\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image039.gif\" width=\"315\" height=\"34\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">Therefore the probability\u00a0that a random student who did not study passed the test is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p6_clip_image042.gif\" width=\"234\" height=\"71\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span>.<\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>Assuming each possibility is equally likely, the <strong><em>probability <\/em><\/strong>of\u00a0a favorable outcome from\u00a0an event is a number between 0 and 1, which measures the\u00a0likelihood that the favorable outcome will occur by using the\u00a0formula:\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p7_clip_image003.gif\" width=\"267\" height=\"37\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/li>\n<li>The <strong><em>sample space <\/em><\/strong>of\u00a0an event is the set of all possible outcomes for that event.<\/li>\n<li>The <em><strong>fundamental\u00a0counting principle <\/strong><\/em>states that the total number of\u00a0possible outcomes from a series of events is determined by\u00a0multiplying all the ways that each individual event can occur.<\/li>\n<li>The <strong><em>permutation<\/em><\/strong> operation <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p7_clip_image006.gif\" width=\"18\" height=\"15\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/> is used to count the elements of the sample space when order<span style=\"text-decoration: none;\">ing <\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\"> elements out of a set of <\/span><em><span style=\"text-decoration: none;\">n<\/span><\/em><span style=\"text-decoration: none;\"> elements and order does matter. <\/span><\/li>\n<li>The <em><strong>combination<\/strong><\/em> operation <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p7_clip_image009.gif\" width=\"21\" height=\"15\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/> is used to count the elements of the sample spa<span style=\"text-decoration: none;\">ce\u00a0when grouping <\/span><em><span style=\"text-decoration: none;\">r <\/span><\/em><span style=\"text-decoration: none;\">elements out of a\u00a0set of <\/span><em><span style=\"text-decoration: none;\">n<\/span><\/em><span style=\"text-decoration: none;\"> elements and order does not matter. <\/span><\/li>\n<li><strong><em>Probability <\/em><\/strong>is the likelihood\u00a0that an event will occur. However, we can look at more than just\u00a0a single event when evaluating probability. <span style=\"text-decoration: none;\">We\u00a0can ask about the probability of two favorable outcomes <\/span><em><span style=\"text-decoration: none;\">both<\/span><\/em><span style=\"text-decoration: none;\"> occurring, and about the probability of <\/span><em><span style=\"text-decoration: none;\">at\u00a0least one<\/span><\/em><span style=\"text-decoration: none;\"> of two\u00a0favorable outcomes occurring. We can even ask about the\u00a0probability of a favorable outcome <\/span><em><span style=\"text-decoration: none;\">not <\/span><\/em><span style=\"text-decoration: none;\">occurring. In all\u00a0of these situations, there are simple formulas defined in terms\u00a0of the known probabilities.<\/span><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p7_clip_image012.gif\" width=\"191\" height=\"18\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s3_p4_clip_image018.gif\" width=\"284\" height=\"24\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p7_clip_image018.gif\" width=\"137\" height=\"18\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><strong><em>Conditional\u00a0probability<\/em><\/strong> meas<span style=\"text-decoration: none;\">ures\u00a0the probability of an outcome <\/span><em><span style=\"text-decoration: none;\">B<\/span><\/em><span style=\"text-decoration: none;\"> provided that we know another outcome <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\"> has already occurred<\/span>.<\/li>\n<li>The formula for conditional probability is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/7\/images\/s4_p7_clip_image021.gif\" width=\"146\" height=\"41\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/utilizing-combinations-and-permutations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/probability-statistics-data-analysis\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/beginning-statistics\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next\u00a0Lesson \u27a1 Independent and Dependent Events Objective In this lesson, you will study the difference between dependent and independent events and how to use conditional\u00a0probability. Previously Covered: Probability is\u00a0the likelihood that an event will occur. What is the difference between independent and dependent\u00a0events? Sometimes the probability\u00a0of an event can change, given the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-297","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/297","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=297"}],"version-history":[{"count":8,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/297\/revisions"}],"predecessor-version":[{"id":702,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/297\/revisions\/702"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}