Q. Two numbers are chosen independently and uniformly at random from the set {1, 2 , . . . , 13}. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary representations have the same most significant bit is<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
The 4-bit binary representation of numbers (1, 2, 3, 4\u2026\u2026\u202613):<\/p>\n\n\n\n
0 – 0000<\/p>\n\n\n\n
1 – 0001<\/p>\n\n\n\n
2 – 0010<\/p>\n\n\n\n
3 – 0011<\/p>\n\n\n\n
4 – 0100<\/p>\n\n\n\n
5 – 0101<\/p>\n\n\n\n
6 – 0110<\/p>\n\n\n\n
7 – 0111<\/p>\n\n\n\n
8 – 1000<\/p>\n\n\n\n
9 – 1001<\/p>\n\n\n\n
10 – 1010<\/p>\n\n\n\n
11 – 1011<\/p>\n\n\n\n
12 – 1100<\/p>\n\n\n\n
13 – 1101<\/p>\n\n\n\n
There 6 numbers which start with MSB as 1, and 7 numbers which start with MSB as 0.<\/p>\n\n\n\n
Therefore, probability that their 4-bit binary representations have the same most significant bit is,<\/p>\n\n\n\n
= P(MSB is 0) + P(MSB is 1)<\/p>\n\n\n\n
= (7\u00d77)\/(13\u00d713) + (6\u00d76)\/(13\u00d713)<\/p>\n\n\n\n
= (49+36)\/169<\/p>\n\n\n\n
= 85\/169<\/p>\n\n\n\n
= 0.5029<\/p>\n","protected":false},"excerpt":{"rendered":"
Q. Two numbers are chosen independently and uniformly at random from the set {1, 2 , . . . , 13}. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary representations have the same most significant bit is Solution: The 4-bit binary representation of numbers (1, 2, 3, 4\u2026\u2026\u202613): 0 – […]<\/p>\n","protected":false},"author":419,"featured_media":154092,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[5141],"tags":[5140],"class_list":["post-154088","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-gate","tag-gate-questions"],"_links":{"self":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts\/154088","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/users\/419"}],"replies":[{"embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/comments?post=154088"}],"version-history":[{"count":1,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts\/154088\/revisions"}],"predecessor-version":[{"id":154093,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts\/154088\/revisions\/154093"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/media\/154092"}],"wp:attachment":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/media?parent=154088"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/categories?post=154088"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/tags?post=154088"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}