{"id":340,"date":"2016-10-11T18:36:08","date_gmt":"2016-10-11T18:36:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/math4libarts\/?post_type=chapter&#038;p=340"},"modified":"2019-05-30T17:01:22","modified_gmt":"2019-05-30T17:01:22","slug":"expected-value","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/expected-value\/","title":{"raw":"Expected Value","rendered":"Expected Value"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Compute a conditional probability for an event<\/li>\r\n \t<li>Use Baye\u2019s theorem to compute a conditional probability<\/li>\r\n \t<li>Calculate the expected value of an event<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Repeating Procedures Over Time<\/h2>\r\nExpected value is perhaps the most useful probability concept we will discuss.\u00a0 It has many applications, from insurance policies to making financial decisions, and it's one thing that the casinos and government agencies that run gambling operations and lotteries hope most people never learn about.<img class=\"aligncenter wp-image-343\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/11182717\/roulette.jpg\" alt=\"A roulette wheel\" width=\"600\" height=\"400\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIn the casino game roulette, a wheel with 38 spaces (18 red, 18 black, and 2 green) is spun. In one possible bet, the player bets $1 on a single number. If that number is spun on the wheel, then they receive $36 (their original $1 + $35). Otherwise, they lose their $1. On average, how much money should a player expect to win or lose if they play this game repeatedly?\r\n[reveal-answer q=\"408866\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"408866\"]\r\n\r\nSuppose you bet $1 on each of the 38 spaces on the wheel, for a total of $38 bet. When the winning number is spun, you are paid $36 on that number. While you won on that one number, overall you\u2019ve lost $2. On a per-space basis, you have \u201cwon\u201d -$2\/$38 \u2248 -$0.053. In other words, on average you lose 5.3 cents per space you bet on.\r\n\r\nWe call this average gain or loss the expected value of playing roulette. Notice that no one ever loses exactly 5.3 cents: most people (in fact, about 37 out of every 38) lose $1 and a very few people (about 1 person out of every 38) gain $35 (the $36 they win minus the $1 they spent to play the game).\r\n\r\nThere is another way to compute expected value without imagining what would happen if we play every possible space.\u00a0 There are 38 possible outcomes when the wheel spins, so the probability of winning is [latex]\\frac{1}{38}[\/latex]. The complement, the probability of losing, is [latex]\\frac{37}{38}[\/latex].\r\n\r\nSummarizing these along with the values, we get this table:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Outcome<\/td>\r\n<td>Probability of outcome<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>$35<\/td>\r\n<td>[latex]\\frac{1}{38}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-$1<\/td>\r\n<td>[latex]\\frac{37}{38}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that if we multiply each outcome by its corresponding probability we get [latex]\\$35\\cdot \\frac{1}{38}=0.9211[\/latex] and [latex]-\\$1\\cdot \\frac{37}{38}=-0.9737[\/latex], and if we add these numbers we get\r\n\r\n0.9211 + (-0.9737) \u2248 -0.053, which is the expected value we computed above.[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Expected Value<\/h3>\r\n<ul>\r\n \t<li><strong>Expected Value<\/strong> is the average gain or loss of an event if the procedure is repeated many times.<\/li>\r\n<\/ul>\r\nWe can compute the expected value by multiplying each outcome by the probability of that outcome, then adding up the products.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nYou purchase a raffle ticket to help out a charity. The raffle ticket costs $5. The charity is selling 2000 tickets. One of them will be drawn and the person holding the ticket will be given a prize worth $4000. Compute the expected value for this raffle.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIn a certain state's lottery, 48 balls numbered 1 through 48 are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. If they match 5 numbers, then win $1,000.\u00a0\u00a0 It costs $1 to buy a ticket. Find the expected value.\r\n[reveal-answer q=\"737029\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"737029\"]\r\n\r\nEarlier, we calculated the probability of matching all 6 numbers and the probability of matching 5 numbers:\r\n\r\n[latex]\\frac{{}_{6}{{C}_{6}}}{{}_{48}{{C}_{6}}}=\\frac{1}{12271512}\\approx0.0000000815[\/latex] for all 6 numbers,\r\n\r\n[latex]\\frac{\\left({}_{6}{{C}_{5}}\\right)\\left({}_{42}{{C}_{1}}\\right)}{{}_{48}{{C}_{6}}}=\\frac{252}{12271512}\\approx0.0000205[\/latex] for 5 numbers.\r\n\r\nOur probabilities and outcome values are:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Outcome<\/td>\r\n<td>Probability of outcome<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>$999,999<\/td>\r\n<td>[latex]\\frac{1}{12271512}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>$999<\/td>\r\n<td>[latex]\\frac{252}{12271512}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-$1<\/td>\r\n<td>[latex]1-\\frac{253}{12271512}=\\frac{12271259}{12271512}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe expected value, then is:\r\n\r\n[latex]\\left(\\$999,999 \\right)\\cdot \\frac{1}{12271512}+\\left( \\$999\\right)\\cdot\\frac{252}{12271512}+\\left(-\\$1\\right)\\cdot\\frac{12271259}{12271512}\\approx-\\$0.898[\/latex]\r\n\r\nOn average, one can expect to lose about 90 cents on a lottery ticket. Of course, most players will lose $1.\r\n\r\n[\/hidden-answer]\r\n\r\nView more about the expected value examples in the following video.\r\n\r\nhttps:\/\/youtu.be\/pFzgxGVltS8\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17430&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"300\"><\/iframe>\r\n\r\n<\/div>\r\nIn general, if the expected value of a game is negative, it is not a good idea to play the game, since on average you will lose money.\u00a0 It would be better to play a game with a positive expected value (good luck trying to find one!), although keep in mind that even if the <em>average<\/em> winnings are positive it could be the case that most people lose money and one very fortunate individual wins a great deal of money.\u00a0 If the expected value of a game is 0, we call it a <strong>fair game<\/strong>, since neither side has an advantage.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA friend offers to play a game, in which you roll 3 standard 6-sided dice. If all the dice roll different values, you give him $1. If any two dice match values, you get $2. What is the expected value of this game? Would you play?\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17431&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nExpected value also has applications outside of gambling. Expected value is very common in making insurance decisions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA 40-year-old man in the U.S. has a 0.242% risk of dying during the next year.[footnote]According to the estimator at <a href=\"http:\/\/www.numericalexample.com\/index.php?view=article&amp;id=91\" target=\"_blank\" rel=\"noopener\">http:\/\/www.numericalexample.com\/index.php?view=article&amp;id=91<\/a>[\/footnote] An insurance company charges $275 for a life-insurance policy that pays a $100,000 death benefit. What is the expected value for the person buying the insurance?\r\n[reveal-answer q=\"90556\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"90556\"]\r\n\r\nThe probabilities and outcomes are\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Outcome<\/td>\r\n<td>Probability of outcome<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>$100,000 - $275 = $99,725<\/td>\r\n<td>0.00242<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-$275<\/td>\r\n<td>1 \u2013 0.00242 = 0.99758<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe expected value is ($99,725)(0.00242) + (-$275)(0.99758) = -$33.\r\n\r\n[\/hidden-answer]\r\n\r\nThe insurance applications of expected value are detailed in the following video.\r\n\r\nhttps:\/\/youtu.be\/Bnai8apt8vw\r\n\r\n<\/div>\r\nNot surprisingly, the expected value is negative; the insurance company can only afford to offer policies if they, on average, make money on each policy. They can afford to pay out the occasional benefit because they offer enough policies that those benefit payouts are balanced by the rest of the insured people.\r\n\r\nFor people buying the insurance, there is a negative expected value, but there is a security that comes from insurance that is worth that cost.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Compute a conditional probability for an event<\/li>\n<li>Use Baye\u2019s theorem to compute a conditional probability<\/li>\n<li>Calculate the expected value of an event<\/li>\n<\/ul>\n<\/div>\n<h2>Repeating Procedures Over Time<\/h2>\n<p>Expected value is perhaps the most useful probability concept we will discuss.\u00a0 It has many applications, from insurance policies to making financial decisions, and it&#8217;s one thing that the casinos and government agencies that run gambling operations and lotteries hope most people never learn about.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-343\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/11182717\/roulette.jpg\" alt=\"A roulette wheel\" width=\"600\" height=\"400\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>In the casino game roulette, a wheel with 38 spaces (18 red, 18 black, and 2 green) is spun. In one possible bet, the player bets $1 on a single number. If that number is spun on the wheel, then they receive $36 (their original $1 + $35). Otherwise, they lose their $1. On average, how much money should a player expect to win or lose if they play this game repeatedly?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q408866\">Show Solution<\/span><\/p>\n<div id=\"q408866\" class=\"hidden-answer\" style=\"display: none\">\n<p>Suppose you bet $1 on each of the 38 spaces on the wheel, for a total of $38 bet. When the winning number is spun, you are paid $36 on that number. While you won on that one number, overall you\u2019ve lost $2. On a per-space basis, you have \u201cwon\u201d -$2\/$38 \u2248 -$0.053. In other words, on average you lose 5.3 cents per space you bet on.<\/p>\n<p>We call this average gain or loss the expected value of playing roulette. Notice that no one ever loses exactly 5.3 cents: most people (in fact, about 37 out of every 38) lose $1 and a very few people (about 1 person out of every 38) gain $35 (the $36 they win minus the $1 they spent to play the game).<\/p>\n<p>There is another way to compute expected value without imagining what would happen if we play every possible space.\u00a0 There are 38 possible outcomes when the wheel spins, so the probability of winning is [latex]\\frac{1}{38}[\/latex]. The complement, the probability of losing, is [latex]\\frac{37}{38}[\/latex].<\/p>\n<p>Summarizing these along with the values, we get this table:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Outcome<\/td>\n<td>Probability of outcome<\/td>\n<\/tr>\n<tr>\n<td>$35<\/td>\n<td>[latex]\\frac{1}{38}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>-$1<\/td>\n<td>[latex]\\frac{37}{38}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that if we multiply each outcome by its corresponding probability we get [latex]\\$35\\cdot \\frac{1}{38}=0.9211[\/latex] and [latex]-\\$1\\cdot \\frac{37}{38}=-0.9737[\/latex], and if we add these numbers we get<\/p>\n<p>0.9211 + (-0.9737) \u2248 -0.053, which is the expected value we computed above.<\/p><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Expected Value<\/h3>\n<ul>\n<li><strong>Expected Value<\/strong> is the average gain or loss of an event if the procedure is repeated many times.<\/li>\n<\/ul>\n<p>We can compute the expected value by multiplying each outcome by the probability of that outcome, then adding up the products.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>You purchase a raffle ticket to help out a charity. The raffle ticket costs $5. The charity is selling 2000 tickets. One of them will be drawn and the person holding the ticket will be given a prize worth $4000. Compute the expected value for this raffle.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>In a certain state&#8217;s lottery, 48 balls numbered 1 through 48 are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. If they match 5 numbers, then win $1,000.\u00a0\u00a0 It costs $1 to buy a ticket. Find the expected value.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q737029\">Show Solution<\/span><\/p>\n<div id=\"q737029\" class=\"hidden-answer\" style=\"display: none\">\n<p>Earlier, we calculated the probability of matching all 6 numbers and the probability of matching 5 numbers:<\/p>\n<p>[latex]\\frac{{}_{6}{{C}_{6}}}{{}_{48}{{C}_{6}}}=\\frac{1}{12271512}\\approx0.0000000815[\/latex] for all 6 numbers,<\/p>\n<p>[latex]\\frac{\\left({}_{6}{{C}_{5}}\\right)\\left({}_{42}{{C}_{1}}\\right)}{{}_{48}{{C}_{6}}}=\\frac{252}{12271512}\\approx0.0000205[\/latex] for 5 numbers.<\/p>\n<p>Our probabilities and outcome values are:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Outcome<\/td>\n<td>Probability of outcome<\/td>\n<\/tr>\n<tr>\n<td>$999,999<\/td>\n<td>[latex]\\frac{1}{12271512}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>$999<\/td>\n<td>[latex]\\frac{252}{12271512}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>-$1<\/td>\n<td>[latex]1-\\frac{253}{12271512}=\\frac{12271259}{12271512}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The expected value, then is:<\/p>\n<p>[latex]\\left(\\$999,999 \\right)\\cdot \\frac{1}{12271512}+\\left( \\$999\\right)\\cdot\\frac{252}{12271512}+\\left(-\\$1\\right)\\cdot\\frac{12271259}{12271512}\\approx-\\$0.898[\/latex]<\/p>\n<p>On average, one can expect to lose about 90 cents on a lottery ticket. Of course, most players will lose $1.<\/p>\n<\/div>\n<\/div>\n<p>View more about the expected value examples in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Expected value\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/pFzgxGVltS8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17430&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<p>In general, if the expected value of a game is negative, it is not a good idea to play the game, since on average you will lose money.\u00a0 It would be better to play a game with a positive expected value (good luck trying to find one!), although keep in mind that even if the <em>average<\/em> winnings are positive it could be the case that most people lose money and one very fortunate individual wins a great deal of money.\u00a0 If the expected value of a game is 0, we call it a <strong>fair game<\/strong>, since neither side has an advantage.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A friend offers to play a game, in which you roll 3 standard 6-sided dice. If all the dice roll different values, you give him $1. If any two dice match values, you get $2. What is the expected value of this game? Would you play?<br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=17431&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>Expected value also has applications outside of gambling. Expected value is very common in making insurance decisions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A 40-year-old man in the U.S. has a 0.242% risk of dying during the next year.<a class=\"footnote\" title=\"According to the estimator at http:\/\/www.numericalexample.com\/index.php?view=article&amp;id=91\" id=\"return-footnote-340-1\" href=\"#footnote-340-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> An insurance company charges $275 for a life-insurance policy that pays a $100,000 death benefit. What is the expected value for the person buying the insurance?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q90556\">Show Solution<\/span><\/p>\n<div id=\"q90556\" class=\"hidden-answer\" style=\"display: none\">\n<p>The probabilities and outcomes are<\/p>\n<table>\n<tbody>\n<tr>\n<td>Outcome<\/td>\n<td>Probability of outcome<\/td>\n<\/tr>\n<tr>\n<td>$100,000 &#8211; $275 = $99,725<\/td>\n<td>0.00242<\/td>\n<\/tr>\n<tr>\n<td>-$275<\/td>\n<td>1 \u2013 0.00242 = 0.99758<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The expected value is ($99,725)(0.00242) + (-$275)(0.99758) = -$33.<\/p>\n<\/div>\n<\/div>\n<p>The insurance applications of expected value are detailed in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Expected value of insurance\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Bnai8apt8vw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>Not surprisingly, the expected value is negative; the insurance company can only afford to offer policies if they, on average, make money on each policy. They can afford to pay out the occasional benefit because they offer enough policies that those benefit payouts are balanced by the rest of the insured people.<\/p>\n<p>For people buying the insurance, there is a negative expected value, but there is a security that comes from insurance that is worth that cost.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-340\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Expected Value. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>Project<\/strong>: Math in Society. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Roulette. <strong>Authored by<\/strong>: Chris Yiu. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.flickr.com\/photos\/clry2\/1366937217\/\">https:\/\/www.flickr.com\/photos\/clry2\/1366937217\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Expected value. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/pFzgxGVltS8\">https:\/\/youtu.be\/pFzgxGVltS8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Expected value of insurance. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Bnai8apt8vw\">https:\/\/youtu.be\/Bnai8apt8vw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-340-1\">According to the estimator at <a href=\"http:\/\/www.numericalexample.com\/index.php?view=article&amp;id=91\" target=\"_blank\" rel=\"noopener\">http:\/\/www.numericalexample.com\/index.php?view=article&amp;id=91<\/a> <a href=\"#return-footnote-340-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":20,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Expected Value\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Roulette\",\"author\":\"Chris Yiu\",\"organization\":\"\",\"url\":\"https:\/\/www.flickr.com\/photos\/clry2\/1366937217\/\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Expected value\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/pFzgxGVltS8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Expected value of insurance\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Bnai8apt8vw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"2d60a362-3c4b-48e0-adac-3af70abbd2be","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-340","chapter","type-chapter","status-publish","hentry"],"part":329,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/340","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/users\/20"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/340\/revisions"}],"predecessor-version":[{"id":3044,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/340\/revisions\/3044"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/parts\/329"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/340\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/media?parent=340"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapter-type?post=340"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/contributor?post=340"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/license?post=340"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}