{"id":26750,"date":"2021-05-12T13:00:39","date_gmt":"2021-05-12T13:00:39","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26750"},"modified":"2024-05-22T00:01:34","modified_gmt":"2024-05-22T00:01:34","slug":"bernoulli-trial-and-binomial-distribution","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/bernoulli-trial-and-binomial-distribution\/","title":{"rendered":"The Bernoulli Trial and the Binomial Distribution"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Bernoulli Trial and Binomial Distribution<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Abstract<\/strong><br \/><em>In this class, we will study the concept of Bernoulli trials and their implications in probability theory. We begin with a detailed definition of Bernoulli trials, then address the concept of independence between events. After clarifying these ideas, we apply the binomial theorem to understand how repeating a Bernoulli trial produces results with a binomial distribution. Finally, practical exercises are proposed to apply and reinforce these concepts.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nBy the end of this class, students will be able to:\n<\/p>\n<ol>\n<li><strong>Identify<\/strong> the main characteristics of Bernoulli trials, including the independence between attempts.<\/li>\n<li><strong>Correctly apply<\/strong> the notation for binomial events derived from Bernoulli trials.<\/li>\n<li><strong>Distinguish<\/strong> between different forms of independence (2-independence, 3-independence, n-independence) and understand their relationship and application in Bernoulli trials.<\/li>\n<li><strong>Understand<\/strong> the relationship between the Bernoulli trial and the binomial theorem, and how this relationship can be used to calculate the probability of a series of successes and failures.<\/li>\n<li><strong>Apply<\/strong> the binomial (or Bernoulli) distribution to calculate the probability of a certain number of successes in a series of attempts.<\/li>\n<\/ol>\n<p><center><br \/>\n<strong><u>TABLE OF CONTENTS<\/u>:<\/strong><br \/>\n<a href=\"#1\">The Bernoulli Trial<\/a><br \/>\n<a href=\"#2\">Different Forms of Independence<\/a><br \/>\n<a href=\"#3\">The Bernoulli Trial and the Binomial Theorem<\/a><br \/>\n<a href=\"#4\">The Binomial (or Bernoulli) Distribution and Probability Distributions<\/a><br \/>\n<a href=\"#5\">Exercises:<\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/uyljDy0bcvU\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>The Bernoulli Trial<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=uyljDy0bcvU&amp;t=103s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A Bernoulli trial is a random experiment<\/span><\/strong><\/a> with a certain probability of success <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">p.<\/span><\/span> If a Bernoulli trial is repeated <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> times identically and independently, we obtain <strong>Bernoulli events:<\/strong> A certain number <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> of successes out of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> attempts. These are also known as <strong>binomial events<\/strong> and are represented using the notation<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Large \\displaystyle Bi(n;k;p)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Another important feature of Bernoulli trials is that all attempts are independent of each other.<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\">EXAMPLE:<\/span> A 6-sided die is repeatedly rolled. Examples of Bernoulli-type events for this experiment are:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><strong>Getting 3 aces out of 5 attempts:<\/strong> represented by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Bi(5;3;1\/6)<\/span><\/span><\/li>\n<li><strong>Getting 7 even numbers out of 12 attempts:<\/strong> represented by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Bi(12;7;1\/3)<\/span><\/span><\/li>\n<\/ul>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h3>Different Forms of Independence<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=uyljDy0bcvU&amp;t=283s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The independence between attempts in a Bernoulli trial<\/span><\/strong><\/a> is not exactly the same independence we have already reviewed; it is a much more restricted version. To explain this difference, let&#8217;s examine the types of independence between events<\/p>\n<h4>2-independence<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/toposuranos.com\/probabilidad-condicional-e-independencia-entre-eventos\/\" rel=\"noopener\" target=\"_blank\">The independence we already know<\/a> is the one between two events. We call it \u00ab2-independence\u00bb. In these terms, we say that events <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> are 2-independent if<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cap B) = P(A)P(B)<\/span><\/span><\/p>\n<h4>3-independence<\/h4>\n<p style=\"text-align: justify; color: #000000;\">Similarly, 3-independence between three events <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C<\/span><\/span> is defined through the relationship<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cap B\\cap C) = P(A)P(B)P(C)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">It is important to highlight that 2-independence between <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C<\/span><\/span> does not necessarily imply 3-independence, although the converse is true.<\/p>\n<h4>n-independence in Bernoulli Trials<\/h4>\n<p style=\"text-align: justify; color: #000000;\">Proceeding similarly to the previous definitions, n-independence between a collection of events <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_1, \\cdots, A_n<\/span><\/span> is defined through the relationship<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Large \\displaystyle P\\left(\\bigcap_{i=1}^n A_i\\right) = \\prod_{i=1}^n P(A_i) <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">And similarly, we have that:<\/p>\n<table style=\"color: #000000;\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(n-1)<\/span><\/span>-independence does not necessarily imply <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span>-independence<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span>-independence <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Longrightarrow<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">(n-1)<\/span><\/span>-independence<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">The <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> repetitions performed in the Bernoulli trial are <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span>-independent.<\/p>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h2>The Bernoulli Trial and the Binomial Theorem<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=uyljDy0bcvU&amp;t=478s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Consider a success and failure experiment<\/span><\/strong><\/a> with a probability of success <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">p<\/span><\/span>; in each attempt, there will consequently be a probability <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1-p<\/span><\/span> of failure. It is clear that the probability of either a success or a failure occurring in each attempt is 1; and since all attempts are independent, the probability of either success or failure occurring in the <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> attempts will be <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1^n.<\/span><\/span> From this, we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Large \\displaystyle 1 = 1^n = [p + (1-p)]^n = \\sum_{k=0}^n {{n}\\choose{k}} p^k(1-p)^{n-k} <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">In the last equality, the <a href=\"https:\/\/toposuranos.com\/ejercicios-de-induccion-matematica-teorema-del-binomio-de-newton\/\" rel=\"noopener\" target=\"_blank\">Newton&#8217;s Binomial Theorem<\/a> has been applied, and the terms within the summation can be interpreted as follows:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle {{n}\\choose{k}}<\/span><\/span>: the number of ways in which <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> successes can occur out of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> attempts<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">p^k<\/span><\/span>: The probability of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> independent successes occurring<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1-p)^{n-k}<\/span><\/span>: The probability of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n-k<\/span><\/span> independent failures occurring<\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">By combining these elements as they appear in the sum, we obtain: the probability of getting <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> successes out of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> attempts; or equivalently, the probability of getting <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n-k<\/span><\/span> failures out of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> attempts.<\/p>\n<p style=\"text-align: justify; color: #000000;\">If we separate each term of the sum, we have the probabilities of getting:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle {{n}\\choose{0}} p^0(1-p)^{n-0} = (1-p)^n<\/span><\/span><\/td>\n<td>0 successes out of n attempts<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle {{n}\\choose{1}} p^1(1-p)^{n-1} = n p(1-p)^{n-1}<\/span><\/span><\/td>\n<td>1 success out of n attempts<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle {{n}\\choose{2}} p^2(1-p)^{n-2}<\/span><\/span><\/td>\n<td>2 successes out of n attempts<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle {{n}\\choose{k}} p^k(1-p)^{n-k}<\/span><\/span><\/td>\n<td>k successes out of n attempts<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle {{n}\\choose{n-1}} p^{n-1}(1-p)^{n-(n-1)} = n p^{n-1}(1-p)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n-1<\/span><\/span> successes out of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> attempts<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle {{n}\\choose{n}} p^{n}(1-p)^{0} = p^{n}<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> successes out of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> attempts<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">And the sum of all these, as we have already seen, is \u00ab1\u00bb. Showing that all possibilities have been covered.<\/p>\n<p style=\"text-align: justify; color: #000000;\">From this, the <strong>probability of the Bernoulli event<\/strong> is defined:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\Large \\color{blue}{P(Bi(n;k;p)) = {{n}\\choose{k}}p^k(1-p)^{n-k}}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Or we also say that <strong>the number of successes <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> has a binomial distribution:<\/strong><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{\\Large \\displaystyle X\\sim Bi(n;p) \\longmapsto P(X=x) = {{n}\\choose{x}}p^x(1-p)^{n-x}}<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>The Binomial (or Bernoulli) Distribution and Probability Distributions<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=uyljDy0bcvU&amp;t=779s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Through the binomial distribution<\/span><\/strong><\/a>, we begin to have the first notions of probability distributions and random variables. In this case, the (discrete) random variable is associated with the number of successes, and its probability distribution is given by the terms of the binomial theorem<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{\\Large \\displaystyle P(X=x) = {{n}\\choose{x}}p^x(1-p)^{n-x}}<\/span><\/span><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/36s4rcfQn7M\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"5\"><\/a><\/br><\/br><\/p>\n<h2>Exercises:<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>A fair 6-sided die is rolled 5 times. Calculate the probability of getting an even number 3 times.<\/li>\n<li>A coin is tossed 10 times. Calculate the probability of getting 0 to 10 heads and make a graph showing the probability for each outcome. How will the graph look if the number of tosses is increased, and the probability of getting a number of heads from 0 to that number of tosses is examined? An Excel spreadsheet may be useful here.<\/li>\n<li>A raffle drum has a quantity <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">s<\/span><\/span> of balls, where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> are gold and the rest are white. All are mixed and one is drawn at random, winning when a gold one is drawn. If this experiment is repeated identically <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">20<\/span><\/span> times, estimate the most probable number of wins for each possible value of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\leq r\\leq s.<\/span><\/span> An Excel spreadsheet may also be useful here.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Bernoulli Trial and Binomial Distribution AbstractIn this class, we will study the concept of Bernoulli trials and their implications in probability theory. We begin with a detailed definition of Bernoulli trials, then address the concept of independence between events. After clarifying these ideas, we apply the binomial theorem to understand how repeating a Bernoulli trial [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26419,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":12,"footnotes":""},"categories":[567,670],"tags":[],"class_list":["post-26750","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-probabilities-and-statistics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The Bernoulli Trial and the Binomial Distribution - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover how repeating a Bernoulli trial produces results with a Binomial Probability Distribution.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/en\/bernoulli-trial-and-binomial-distribution\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Bernoulli Trial and the Binomial Distribution\" \/>\n<meta property=\"og:description\" content=\"Discover how repeating a Bernoulli trial produces results with a Binomial Probability Distribution.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/en\/bernoulli-trial-and-binomial-distribution\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-05-12T13:00:39+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-22T00:01:34+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/04\/bernoulli-e1712960368446-1024x285.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"The Bernoulli Trial and the Binomial Distribution\" \/>\n<meta name=\"twitter:description\" content=\"Discover how repeating a Bernoulli trial produces results with a Binomial Probability Distribution.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/04\/bernoulli-e1712960368446.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/bernoulli-trial-and-binomial-distribution\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/bernoulli-trial-and-binomial-distribution\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"The Bernoulli Trial and the Binomial Distribution\",\"datePublished\":\"2021-05-12T13:00:39+00:00\",\"dateModified\":\"2024-05-22T00:01:34+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/bernoulli-trial-and-binomial-distribution\\\/\"},\"wordCount\":1215,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/bernoulli-trial-and-binomial-distribution\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/04\\\/bernoulli-e1712960368446.jpg\",\"articleSection\":[\"Mathematics\",\"Probabilities and Statistics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/bernoulli-trial-and-binomial-distribution\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/bernoulli-trial-and-binomial-distribution\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/bernoulli-trial-and-binomial-distribution\\\/\",\"name\":\"The Bernoulli Trial and the Binomial Distribution - 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