{"id":20335,"date":"2024-04-15T05:51:30","date_gmt":"2024-04-15T05:51:30","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20335"},"modified":"2024-04-15T05:51:30","modified_gmt":"2024-04-15T05:51:30","slug":"for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/","title":{"rendered":"For each element in a set of size 2n, an unbiased coin is tossed. All the 2n coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is A. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{4^{\\text{n}}}}}\\] B. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}\\] C. \\[\\frac{1}{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}\\] D. $$\\frac{1}{2}$$"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a9817e3ac83f\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    &#8221;[\frac{{left(                <\/div>\r\n                                                                                                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    &#8221; option2=&#8221;\\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}\\]&#8221; option3=&#8221;\\[\\frac{1}{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}\\]&#8221; option4=&#8221;$$\\frac{1}{2}$$&#8221; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}}$.<\/p>\n<p>To see why, let&#8217;s consider the following example. Suppose we have a set of 4 elements, $A$, $B$, $C$, and $D$, and we toss a coin for each element. There are $2^4 = 16$ possible outcomes, corresponding to all the possible combinations of heads and tails. Of these outcomes, only 4 of them (HHHT, HTHH, HTH, and THHH) will result in exactly 2 heads being chosen.<\/p>\n<p>In general, for a set of $2n$ elements, there are $2^{2n}$ possible outcomes. Of these, <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> there are $\\binom{2n}{n}$ outcomes that will result in exactly $n$ heads being chosen. Therefore, the probability of exactly $n$ heads being chosen is $\\frac{\\binom{2n}{n}}{2^{2n}}$.<\/p>\n<p>We can also calculate this probability using a more formal argument. Let $X$ be the number of heads that are chosen. Then $X$ is a binomial random variable with parameters $n$ and $p = \\frac{1}{2}$. The probability that $X = n$ is therefore given by<\/p>\n<p>$$P(X = n) = \\binom{2n}{n} \\left( \\frac{1}{2} \\right)^n \\left( \\frac{1}{2} \\right)^{2n-n} = \\frac{\\binom{2n}{n}}{2^{2n}}$$<\/p>\n<p>Therefore, the probability that exactly $n$ elements are chosen is $\\boxed{\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}}$.<\/p>\n<p>The other options <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> are incorrect. Option A is the probability that all $2n$ elements are chosen. Option B is the probability that none of the elements are chosen. Option C is the probability that at least one element is chosen.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#8221; option2=&#8221;\\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}\\]&#8221; option3=&#8221;\\[\\frac{1}{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}\\]&#8221; Subscribe on YouTube option4=&#8221;$$\\frac{1}{2}$$&#8221; correct=&#8221;option1&#8243;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[691],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>For each element in a set of size 2n, an unbiased coin is tossed. All the 2n coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is A. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{4^{\\text{n}}}}}\\] B. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}\\] C. \\[\\frac{1}{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}\\] D. $$\\frac{1}{2}$$<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"For each element in a set of size 2n, an unbiased coin is tossed. All the 2n coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is A. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{4^{\\text{n}}}}}\\] B. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}\\] C. \\[\\frac{1}{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}\\] D. $$\\frac{1}{2}$$\" \/>\n<meta property=\"og:description\" content=\"&#8221; option2=&#8221;[frac{{left( {begin{array}{*{20}{c}} {2{text{n}}} \\ {text{n}} end{array}} right)}}{{{2^{text{n}}}}}]&#8221; option3=&#8221;[frac{1}{{left( {begin{array}{*{20}{c}} Subscribe on YouTube {2{text{n}}} \\ {text{n}} end{array}} right)}}]&#8221; option4=&#8221;$$frac{1}{2}$$&#8221; correct=&#8221;option1&#8243;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:51:30+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"For each element in a set of size 2n, an unbiased coin is tossed. All the 2n coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is A. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{4^{\\text{n}}}}}\\] B. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}\\] C. \\[\\frac{1}{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}\\] D. $$\\frac{1}{2}$$","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/","og_locale":"en_US","og_type":"article","og_title":"For each element in a set of size 2n, an unbiased coin is tossed. All the 2n coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is A. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{4^{\\text{n}}}}}\\] B. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}\\] C. \\[\\frac{1}{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}\\] D. $$\\frac{1}{2}$$","og_description":"&#8221; option2=&#8221;[frac{{left( {begin{array}{*{20}{c}} {2{text{n}}} \\ {text{n}} end{array}} right)}}{{{2^{text{n}}}}}]&#8221; option3=&#8221;[frac{1}{{left( {begin{array}{*{20}{c}} Subscribe on YouTube {2{text{n}}} \\ {text{n}} end{array}} right)}}]&#8221; option4=&#8221;$$frac{1}{2}$$&#8221; correct=&#8221;option1&#8243;]","og_url":"https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:51:30+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/","url":"https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/","name":"For each element in a set of size 2n, an unbiased coin is tossed. All the 2n coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is A. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{4^{\\text{n}}}}}\\] B. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}\\] C. \\[\\frac{1}{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}\\] D. $$\\frac{1}{2}$$","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:51:30+00:00","dateModified":"2024-04-15T05:51:30+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/for-each-element-in-a-set-of-size-2n-an-unbiased-coin-is-tossed-all-the-2n-coin-tossed-are-independent-an-element-is-chosen-if-the-corresponding-coin-toss-were-head-the-probability-that-exactly-n\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"mcq","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/"},{"@type":"ListItem","position":3,"name":"Engineering maths","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/"},{"@type":"ListItem","position":4,"name":"Probability and statistics","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/engineering-maths\/probability-and-statistics\/"},{"@type":"ListItem","position":5,"name":"For each element in a set of size 2n, an unbiased coin is tossed. All the 2n coin tossed are independent. An element is chosen if the corresponding coin toss were head. The probability that exactly n elements are chosen is A. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{4^{\\text{n}}}}}\\] B. \\[\\frac{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}{{{2^{\\text{n}}}}}\\] C. \\[\\frac{1}{{\\left( {\\begin{array}{*{20}{c}} {2{\\text{n}}} \\\\ {\\text{n}} \\end{array}} \\right)}}\\] D. $$\\frac{1}{2}$$"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/d97f17072bfa490596c8f78363955d55?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/d97f17072bfa490596c8f78363955d55?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20335"}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=20335"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20335\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=20335"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=20335"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=20335"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}