{"id":7138,"date":"2022-01-31T13:40:03","date_gmt":"2022-01-31T08:10:03","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7138"},"modified":"2025-11-10T17:31:07","modified_gmt":"2025-11-10T12:01:07","slug":"what-is-binomial-distribution","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-binomial-distribution\/","title":{"rendered":"What is binomial distribution?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\" id=\"introduction\">Introduction<\/h2>\n\n\n\n<p>A distribution is said to be a binomial distribution if the following conditions are met.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Each trial has a binary outcome (One of the two outcomes is labeled a \u2018success\u2019)<\/li>\n\n\n\n<li>The probability of success is known and constant over all trials<\/li>\n\n\n\n<li>The number of trials is specified<\/li>\n\n\n\n<li>The trials are independent. That is, the outcome from one trial doesn\u2019t affect the outcome of successive trials<\/li>\n<\/ol>\n\n\n\n<p>If all the above conditions met then the binomial distribution describes the probability of X successes in n trials.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"binomial-distribution\">Binomial distribution<\/h2>\n\n\n\n<p>A classic example of the binomial distribution is the number of heads (X) in n coin tosses.<img loading=\"lazy\" decoding=\"async\" width=\"158\" height=\"41\" src=\"https:\/\/lh5.googleusercontent.com\/fVzrbBObYy8E8a1PEacTI-UYa_CQX-ix0Tkm9HMnKGQ47KpZvfyCJy7EiWCrJAAmnE8QHGW4QCOCB5pXzjGWTYdZPXy-knu6SAjhQtGQsMtzYGxQIEzzFQKQerCCTx-ScQbQX-QF\"><\/p>\n\n\n\n<p>To find the probability of the happening of an event once, twice, thrice,\u2026r times \u2026.exactly in n trails.<\/p>\n\n\n\n<p>Let the probability of the happening of an event A in one trial be p and its probability of not happening be 1 \u2013 p \u2013 q.<\/p>\n\n\n\n<p>We assume that there are n trials and the happening of the event A is r times and its not happening is n \u2013 r times.<\/p>\n\n\n\n<p>This may be shown as follows<\/p>\n\n\n\n<p>AA\u2026\u2026A &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/MJtWvZ_XBJ283v26J9UXKdRhD-AaXbXEE_cy4YBhOmFVXNryjYSuhorypp4tpmUxI28f4sEsl5FPI0mJ7h4cKBdH1fIqTyKizu3pQy2xFPaUK573xbz3fZTzGGcKPCJPTdMvqOf6\" width=\"94\" height=\"31\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p>r times &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; n \u2013 r times&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (1)<\/p>\n\n\n\n<p>A indicates its happening <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/EJsddb5Gn2f1bIDKnCgVugJPjlHMQyn-1nXVHxQJ_3Mqm299i5dBjNpjYVnLt3qVEYO9FUXQived1-r5B8JclclKJbr9YYoVCUnACRUoCxUGiyueG7mzJu6DubSiBsP_U3xNKx5B\" width=\"17\" height=\"36\"> its failure and P (A) =p and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/5gaY_YRKduo4tzVV-O0SI4z4WtpyMFFq_wburVFHdpCJ9XjZ2hQiyeJ2kDeG3bfabDSPJSarXczuVZYyl38To73Rd70FuSDeOaMEfPZGf8kT2BDWPZOmbs4u6m65ng9rcQ6P-6vZ\" width=\"69\" height=\"29\"><\/p>\n\n\n\n<p>We see that (1) has the probability<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/MhZdo3tQjIxCt2YJghFtPqCQCWfJomNtZe6R3l1q4lxmzU3XXYT7yMCNBx64QOfXD7POCah3uh6Zg3fsXvhoOFC4-wB6d2ECUhGC4IxZe-S_N5dvR4pEE3HewXmXZm3EVtpwYJxu\" alt=\"\"\/><\/figure>\n\n\n\n<p>r times&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; n-r times &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (2)<\/p>\n\n\n\n<p>Clearly (1) is merely one order of arranging r A\u2019S.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/HyVW3tKR_2VaagTrBOv3q0iOlTRFGvW507LeL5YZqTHsZMZ0YrMXTkF3ejhzOoCsqn2cnO81NETBZjqeM_CUG9cCugND3xm29PpBoB5s1nP5Y8kv46ReSNVI9NLDbdO6XzyDunhU\" alt=\"\"\/><\/figure>\n\n\n\n<p>These terms are clearly the successive terms in the expansion of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/VbhHhJiCRoHexdeWzjWGKgekTkp8-DiMhsnCGy50YIsTCgdggiQETQx3mpi4ZqMshMqbRSCmdh-LhAi2wItPuFKz5lihVq1EEZzhvwrZOuDpKxkfhcH-vPRF7oBvIeHoffo2HSSl\" width=\"68\" height=\"23\"><\/p>\n\n\n\n<p>Hence it is called Binomial Distribution.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"solved-examples\">Solved examples<\/h2>\n\n\n\n<p><strong>Example. If on an average one ship in every ten is wrecked. Find the probability that out of 5 ships expected to arrive, 4 at least we will arrive safely.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Out of 10 ships one ship is wrecked.<\/p>\n\n\n\n<p>I.e. nine ships out of 10 ships are safe, P (safety) = 9\/10<\/p>\n\n\n\n<p>P (at least 4 ships out of 5 are safe) = P (4 or 5) = P (4) + P(5)<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/cOs5Ekdj5Zej9UM9ufxlP3bQZjNRWxHFxG6xHiR1uitefCP_IZs0w4xNdzzxAuwCRjJNqO_nriVvkFjQOK3Lx-ZUwwYTeAtTYu62g3zItlm-9C14NEo9ut4BwRobmttR3nCNbO5z\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Example. The overall percentage of failures in a certain examination is 20. if 6 candidates appear in the examination what is the probability that at least five pass the examination?<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Probability of failures = 20% = 20\/100 = 1\/5<\/p>\n\n\n\n<p>(P) = 1- \u2155 = 4\/5<\/p>\n\n\n\n<p>Probability of at least 5 pass = P(5 or 6)<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/QDN9zQqS9yZ0Y84Il5G9zUBtMs2uQuTxTJ5PEVM7Qkb79FMmW5aayYmg1aQhWGoROMz_nnngn8_wTCKu7mo6gs7A_wqH9WD78DbuwnRzl2PDf6VCGIuZdpkIBgUmXJFTdeIaUXOO\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Example:<\/strong><\/p>\n\n\n\n<p><strong>The probability that a man aged 60 will live to be 70 is 0.65. what is the probability that out of 10 men, now 60, at least seven will live to be 70?<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>The probability that a man aged 60 will live to be 70 = p = 0.65<\/p>\n\n\n\n<p>q = 1 &#8211; 0.65 = 0.35<\/p>\n\n\n\n<p>Number of men= n = 10<\/p>\n\n\n\n<p>Probability that at least 7 men will live to 70 = (7 or 8 or 9 or 10)<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/PXSQvs8DZTaQlkz8agojcyWWTkUEXAp0Xn2qaiVMDnJEQV2NYD--2KMd4jfY3z5eaBIhuTP3sze1tjyCZfv1n_IGKzNMgxIS4iAuQXDGeX4_f8feEnDsASXI-Vz-t1q73BkMP6Ye\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Example<\/strong><\/p>\n\n\n\n<p><strong>Assuming that 20% of the population of a city are literate so that the chance of an individual being literate is 1\/5 and assuming that hundred investigators each take 10 individuals to see whether they are illiterate, how many investigators would you expect to report 3 or less were literate.<a href=\"https:\/\/chickendogfood.com\/can-dogs-eat-lamb-bones\/\" style=\"color: #000000; text-decoration: none;\">https:\/\/chickendogfood.com\/can-dogs-eat-lamb-bones\/<a><\/strong><\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/sgOCCoHbL4myidHaW5ZZEqrE_lSJHUR8eBrp84GAclC4WD7o9EtOb8qK_udsnoGT8lx0xIDcLOvuQcPJ-Kknk8mkQ1SNizOYQFZAQmst-DOHkhDGJ8lzvzdRgpPIFjmSN51i9Q9h\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Required number of investigators = 0.879126118\u00d7 100 =87.9126118 =&nbsp; = 88 approximate<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A distribution is said to be a binomial distribution if the following conditions are met. 1. Each trial has a binary outcome (One of the two outcomes is labeled a \u2018success\u2019)<\/p>\n","protected":false},"author":28,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[],"class_list":["post-7138","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is binomial distribution? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"A distribution is said to be a binomial distribution if the following conditions are met. 1. 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