A Bernoulli trial is an experiment that results in a success with probability $p$ and a failure with probability $1-p$. A random variable is said to have a Binomial Distribution if it is the result of recording the number of successes in n independent Bernoulli trials.<\/p>\n
In the Applet below, we have represented repeated independent Bernoulli trials by a single ball falling through an array of pins. Each time a ball falls onto a pin, it will bounce to the right (i.e. a success) with probability $p$ or to the left (i.e. a failure) with probability $1-p$. After the ball falls through the array, it lands in a bin labeled by the corresponding number of successes.<\/p>\n
Click on a bin to see its corresponding total and probability. Alternatively, use the left and right arrow buttons to scroll through the bins. Also displayed is a confidence interval centered on the theoretical expected bin. Bins that are included in this confidence interval are highlighted in green.<\/p>\n
\r\n<\/APPLET><\/p>\n\n\nDavid Little: http:\/\/www.personal.psu.edu\/dpl14\/java\/probability\/plinko\/index.html<\/a><\/p>\n<\/li>\n<\/ul>\nPlinko II >><\/a><\/span><\/div><\/div>\n\n\nPlinko II<\/b><\/span><\/p>\nPlinko Probability<\/h4>\n\r\n<\/iframe><\/p>\n<ul>\n<li>\n<div style=\"text-align: left;\">Plinko Probability: <a href=\"http:\/\/phet.colorado.edu\/pt\/simulation\/plinko-probability\" target=\"_blank\" rel=\"noopener\">http:\/\/phet.colorado.edu\/pt\/simulation\/plinko-probability<\/a><\/div>\n<\/li>\n<\/ul>\n<p style=\"text-align: left;\"><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(0,7158)'><< Plinko I<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(2,7158)'>Galton Box >><\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_7158'>\n<span class='GTTabs_titles'><b>Galton Box<\/b><\/span><\/p>\n<div id=\"attachment_7160\" style=\"width: 230px\" class=\"wp-caption alignleft\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-7160\" data-attachment-id=\"7160\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7160\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg\" data-orig-size=\"400,543\" data-comments-opened=\"1\" data-image-meta=\"{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":""}\" data-image-title=\"Francis Galton\" data-image-description=\"\" data-image-caption=\"<p>Sir Francis Galton<\/p>\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg\" class=\" wp-image-7160 size-medium\" title=\"Francis Galton\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s-220x300.jpg\" alt=\"\" width=\"220\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s-220x300.jpg 220w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s-110x150.jpg 110w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg 400w\" sizes=\"auto, (max-width: 220px) 100vw, 220px\" \/><\/a><p id=\"caption-attachment-7160\" class=\"wp-caption-text\">Sir Francis Galton<\/p><\/div>\n<p>The bean machine, also known as the quincunx or Galton box, is a device invented by Sir <a href=\"http:\/\/en.wikipedia.org\/wiki\/Francis_Galton\" target=\"_blank\" rel=\"noopener\">Francis Galton<\/a> to demonstrate the central limit theorem, in particular that the normal distribution is approximate to the binomial distribution.<\/p>\n<div id=\"attachment_7159\" style=\"width: 360px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-7159\" data-attachment-id=\"7159\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7159\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png\" data-orig-size=\"744,577\" data-comments-opened=\"1\" data-image-meta=\"{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":""}\" data-image-title=\"Quincunx\" data-image-description=\"\" data-image-caption=\"<p>The bean machine, as drawn by Sir Francis Galton<\/p>\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png\" class=\" wp-image-7159\" title=\"Quincunx\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-300x232.png\" alt=\"\" width=\"350\" height=\"271\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-300x232.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-150x116.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-400x310.png 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png 744w\" sizes=\"auto, (max-width: 350px) 100vw, 350px\" \/><\/a><p id=\"caption-attachment-7159\" class=\"wp-caption-text\">The bean machine, as drawn by Sir Francis Galton<\/p><\/div>\n<p>The machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce left and right as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom. The height of ball columns in the bins approximates a bell curve.<\/p>\n<p>Overlaying Pascal’s triangle onto the pins shows the number of different paths that can be taken to get to each bin.<\/p>\n<p>A large-scale working model of this device can be seen at the Museum of Science, Boston in the Mathematica exhibit.<\/p>\n<h5>Distribution of the balls<\/h5>\n<p>If a ball bounces to the right k times on its way down (and to the left on the remaining pins) it ends up in the kth bin counting from the left. Denoting the number of rows of pins in a bean machine by n, the number of paths to the kth bin on the bottom is given by the binomial coefficient ${}^{n}{{C}_{k}}$. If the probability of bouncing right on a pin is p (which equals 0.5 on an unbiased machine) the probability that the ball ends up in the kth bin equals ${}^{n}{{C}_{k}}\\,{{p}^{k}}{{(1-p)}^{n-k}}$. This is the probability mass function of a binomial distribution.<\/p>\n<p>According to the central limit theorem the binomial distribution approximates the normal distribution provided that n, the number of rows of pins in the machine, is large.<\/p>\n<ul>\n<li style=\"text-align: left;\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/Bean_machine\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/Bean_machine<\/a><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"https:\/\/www.youtube-nocookie.com\/embed\/uFd3hiZZHWg?rel=0\" width=\"640\" height=\"480\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"https:\/\/www.youtube-nocookie.com\/embed\/AUSKTk9ENzg?rel=0\" width=\"640\" height=\"480\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p style=\"text-align: left;\"><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(1,7158)'><< Plinko II<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(3,7158)'>Folha de c\u00e1lculo >><\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_3_7158'>\n<span class='GTTabs_titles'><b>Folha de c\u00e1lculo<\/b><\/span><\/p>\n<p>[embeddoc url=”https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-Binon.xls” height=”500px” viewer=”microsoft”]<\/p>\n<ul>\n<li>\n<div style=\"text-align: left;\">\u00a0<a href=\"https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-Binon.xls\">https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-Binon.xls<\/a><\/div>\n<\/li>\n<\/ul>\n<p style=\"text-align: left;\">\u00a0<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(2,7158)'><< Galton Box<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(4,7158)'>GeoGebra >><\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_4_7158'>\n<span class='GTTabs_titles'><b>GeoGebra<\/b><\/span><\/p>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":960,\r\n\"height\":500,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 5 19 67 | 2 15 45 18 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 65 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 | 30 29 54 32 31 33 | 25 , 17 26 62 , 14 66 68 | 52 60 61 || 40 41 42 , 27 28 35 , 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David Little: http:\/\/www.personal.psu.edu\/dpl14\/java\/probability\/plinko\/index.html<\/a><\/p>\n<\/li>\n<\/ul>\nPlinko II >><\/a><\/span><\/div><\/div>\n\n\nPlinko II<\/b><\/span><\/p>\nPlinko Probability<\/h4>\n\r\n<\/iframe><\/p>\n<ul>\n<li>\n<div style=\"text-align: left;\">Plinko Probability: <a href=\"http:\/\/phet.colorado.edu\/pt\/simulation\/plinko-probability\" target=\"_blank\" rel=\"noopener\">http:\/\/phet.colorado.edu\/pt\/simulation\/plinko-probability<\/a><\/div>\n<\/li>\n<\/ul>\n<p style=\"text-align: left;\"><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(0,7158)'><< Plinko I<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(2,7158)'>Galton Box >><\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_7158'>\n<span class='GTTabs_titles'><b>Galton Box<\/b><\/span><\/p>\n<div id=\"attachment_7160\" style=\"width: 230px\" class=\"wp-caption alignleft\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-7160\" data-attachment-id=\"7160\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7160\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg\" data-orig-size=\"400,543\" data-comments-opened=\"1\" data-image-meta=\"{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":""}\" data-image-title=\"Francis Galton\" data-image-description=\"\" data-image-caption=\"<p>Sir Francis Galton<\/p>\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg\" class=\" wp-image-7160 size-medium\" title=\"Francis Galton\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s-220x300.jpg\" alt=\"\" width=\"220\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s-220x300.jpg 220w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s-110x150.jpg 110w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg 400w\" sizes=\"auto, (max-width: 220px) 100vw, 220px\" \/><\/a><p id=\"caption-attachment-7160\" class=\"wp-caption-text\">Sir Francis Galton<\/p><\/div>\n<p>The bean machine, also known as the quincunx or Galton box, is a device invented by Sir <a href=\"http:\/\/en.wikipedia.org\/wiki\/Francis_Galton\" target=\"_blank\" rel=\"noopener\">Francis Galton<\/a> to demonstrate the central limit theorem, in particular that the normal distribution is approximate to the binomial distribution.<\/p>\n<div id=\"attachment_7159\" style=\"width: 360px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-7159\" data-attachment-id=\"7159\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7159\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png\" data-orig-size=\"744,577\" data-comments-opened=\"1\" data-image-meta=\"{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":""}\" data-image-title=\"Quincunx\" data-image-description=\"\" data-image-caption=\"<p>The bean machine, as drawn by Sir Francis Galton<\/p>\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png\" class=\" wp-image-7159\" title=\"Quincunx\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-300x232.png\" alt=\"\" width=\"350\" height=\"271\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-300x232.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-150x116.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-400x310.png 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png 744w\" sizes=\"auto, (max-width: 350px) 100vw, 350px\" \/><\/a><p id=\"caption-attachment-7159\" class=\"wp-caption-text\">The bean machine, as drawn by Sir Francis Galton<\/p><\/div>\n<p>The machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce left and right as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom. The height of ball columns in the bins approximates a bell curve.<\/p>\n<p>Overlaying Pascal’s triangle onto the pins shows the number of different paths that can be taken to get to each bin.<\/p>\n<p>A large-scale working model of this device can be seen at the Museum of Science, Boston in the Mathematica exhibit.<\/p>\n<h5>Distribution of the balls<\/h5>\n<p>If a ball bounces to the right k times on its way down (and to the left on the remaining pins) it ends up in the kth bin counting from the left. Denoting the number of rows of pins in a bean machine by n, the number of paths to the kth bin on the bottom is given by the binomial coefficient ${}^{n}{{C}_{k}}$. If the probability of bouncing right on a pin is p (which equals 0.5 on an unbiased machine) the probability that the ball ends up in the kth bin equals ${}^{n}{{C}_{k}}\\,{{p}^{k}}{{(1-p)}^{n-k}}$. This is the probability mass function of a binomial distribution.<\/p>\n<p>According to the central limit theorem the binomial distribution approximates the normal distribution provided that n, the number of rows of pins in the machine, is large.<\/p>\n<ul>\n<li style=\"text-align: left;\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/Bean_machine\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/Bean_machine<\/a><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"https:\/\/www.youtube-nocookie.com\/embed\/uFd3hiZZHWg?rel=0\" width=\"640\" height=\"480\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"https:\/\/www.youtube-nocookie.com\/embed\/AUSKTk9ENzg?rel=0\" width=\"640\" height=\"480\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p style=\"text-align: left;\"><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(1,7158)'><< Plinko II<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(3,7158)'>Folha de c\u00e1lculo >><\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_3_7158'>\n<span class='GTTabs_titles'><b>Folha de c\u00e1lculo<\/b><\/span><\/p>\n<p>[embeddoc url=”https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-Binon.xls” height=”500px” viewer=”microsoft”]<\/p>\n<ul>\n<li>\n<div style=\"text-align: left;\">\u00a0<a href=\"https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-Binon.xls\">https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-Binon.xls<\/a><\/div>\n<\/li>\n<\/ul>\n<p style=\"text-align: left;\">\u00a0<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(2,7158)'><< Galton Box<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(4,7158)'>GeoGebra >><\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_4_7158'>\n<span class='GTTabs_titles'><b>GeoGebra<\/b><\/span><\/p>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":960,\r\n\"height\":500,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 5 19 67 | 2 15 45 18 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 65 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 | 30 29 54 32 31 33 | 25 , 17 26 62 , 14 66 68 | 52 60 61 || 40 41 42 , 27 28 35 , 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A random variable...<\/p>\n","protected":false},"author":1,"featured_media":21027,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,251,252],"series":[],"class_list":["post-7158","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-probabilidades-e-combinatoria","tag-12-o-ano","tag-distribuicao-binomial","tag-galton"],"views":2486,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/12-Distribuicao_Binomial.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7158","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7158"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7158\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21027"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7158"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7158"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7158"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\r\n<\/iframe><\/p>\n<ul>\n<li>\n<div style=\"text-align: left;\">Plinko Probability: <a href=\"http:\/\/phet.colorado.edu\/pt\/simulation\/plinko-probability\" target=\"_blank\" rel=\"noopener\">http:\/\/phet.colorado.edu\/pt\/simulation\/plinko-probability<\/a><\/div>\n<\/li>\n<\/ul>\n<p style=\"text-align: left;\"><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(0,7158)'><< Plinko I<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(2,7158)'>Galton Box >><\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_7158'>\n<span class='GTTabs_titles'><b>Galton Box<\/b><\/span><\/p>\n<div id=\"attachment_7160\" style=\"width: 230px\" class=\"wp-caption alignleft\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-7160\" data-attachment-id=\"7160\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7160\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg\" data-orig-size=\"400,543\" data-comments-opened=\"1\" data-image-meta=\"{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":""}\" data-image-title=\"Francis Galton\" data-image-description=\"\" data-image-caption=\"<p>Sir Francis Galton<\/p>\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg\" class=\" wp-image-7160 size-medium\" title=\"Francis Galton\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s-220x300.jpg\" alt=\"\" width=\"220\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s-220x300.jpg 220w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s-110x150.jpg 110w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Francis_Galton_1850s.jpg 400w\" sizes=\"auto, (max-width: 220px) 100vw, 220px\" \/><\/a><p id=\"caption-attachment-7160\" class=\"wp-caption-text\">Sir Francis Galton<\/p><\/div>\n<p>The bean machine, also known as the quincunx or Galton box, is a device invented by Sir <a href=\"http:\/\/en.wikipedia.org\/wiki\/Francis_Galton\" target=\"_blank\" rel=\"noopener\">Francis Galton<\/a> to demonstrate the central limit theorem, in particular that the normal distribution is approximate to the binomial distribution.<\/p>\n<div id=\"attachment_7159\" style=\"width: 360px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-7159\" data-attachment-id=\"7159\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7159\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png\" data-orig-size=\"744,577\" data-comments-opened=\"1\" data-image-meta=\"{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":""}\" data-image-title=\"Quincunx\" data-image-description=\"\" data-image-caption=\"<p>The bean machine, as drawn by Sir Francis Galton<\/p>\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png\" class=\" wp-image-7159\" title=\"Quincunx\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-300x232.png\" alt=\"\" width=\"350\" height=\"271\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-300x232.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-150x116.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram-400x310.png 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/Quincunx_Galton_Box_-_Galton_1889_diagram.png 744w\" sizes=\"auto, (max-width: 350px) 100vw, 350px\" \/><\/a><p id=\"caption-attachment-7159\" class=\"wp-caption-text\">The bean machine, as drawn by Sir Francis Galton<\/p><\/div>\n<p>The machine consists of a vertical board with interleaved rows of pins. Balls are dropped from the top, and bounce left and right as they hit the pins. Eventually, they are collected into one-ball-wide bins at the bottom. The height of ball columns in the bins approximates a bell curve.<\/p>\n<p>Overlaying Pascal’s triangle onto the pins shows the number of different paths that can be taken to get to each bin.<\/p>\n<p>A large-scale working model of this device can be seen at the Museum of Science, Boston in the Mathematica exhibit.<\/p>\n<h5>Distribution of the balls<\/h5>\n<p>If a ball bounces to the right k times on its way down (and to the left on the remaining pins) it ends up in the kth bin counting from the left. Denoting the number of rows of pins in a bean machine by n, the number of paths to the kth bin on the bottom is given by the binomial coefficient ${}^{n}{{C}_{k}}$. If the probability of bouncing right on a pin is p (which equals 0.5 on an unbiased machine) the probability that the ball ends up in the kth bin equals ${}^{n}{{C}_{k}}\\,{{p}^{k}}{{(1-p)}^{n-k}}$. This is the probability mass function of a binomial distribution.<\/p>\n<p>According to the central limit theorem the binomial distribution approximates the normal distribution provided that n, the number of rows of pins in the machine, is large.<\/p>\n<ul>\n<li style=\"text-align: left;\"><a href=\"http:\/\/en.wikipedia.org\/wiki\/Bean_machine\" target=\"_blank\" rel=\"noopener\">http:\/\/en.wikipedia.org\/wiki\/Bean_machine<\/a><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"https:\/\/www.youtube-nocookie.com\/embed\/uFd3hiZZHWg?rel=0\" width=\"640\" height=\"480\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"https:\/\/www.youtube-nocookie.com\/embed\/AUSKTk9ENzg?rel=0\" width=\"640\" height=\"480\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p style=\"text-align: left;\"><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(1,7158)'><< Plinko II<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(3,7158)'>Folha de c\u00e1lculo >><\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_3_7158'>\n<span class='GTTabs_titles'><b>Folha de c\u00e1lculo<\/b><\/span><\/p>\n<p>[embeddoc url=”https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-Binon.xls” height=”500px” viewer=”microsoft”]<\/p>\n<ul>\n<li>\n<div style=\"text-align: left;\">\u00a0<a href=\"https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-Binon.xls\">https:\/\/www.acasinhadamatematica.pt\/cm\/recursos_materiais\/alabmat\/0_ficheiros\/Dist-Binon.xls<\/a><\/div>\n<\/li>\n<\/ul>\n<p style=\"text-align: left;\">\u00a0<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(2,7158)'><< Galton Box<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7158' onClick='GTTabs_show(4,7158)'>GeoGebra >><\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_4_7158'>\n<span class='GTTabs_titles'><b>GeoGebra<\/b><\/span><\/p>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":960,\r\n\"height\":500,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 5 19 67 | 2 15 45 18 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 65 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 | 30 29 54 32 31 33 | 25 , 17 26 62 , 14 66 68 | 52 60 61 || 40 41 42 , 27 28 35 , 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