{"id":188798,"date":"2025-02-07T09:51:40","date_gmt":"2025-02-07T09:51:40","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=188798"},"modified":"2025-02-07T09:51:42","modified_gmt":"2025-02-07T09:51:42","slug":"the-probability-of-the-union-of-two-events-with-nonzero-probabilities","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/07\/the-probability-of-the-union-of-two-events-with-nonzero-probabilities\/","title":{"rendered":"The probability of the union of two events with nonzero probabilities"},"content":{"rendered":"\n

The probability of the union of two events with nonzero probabilities<\/p>\n\n\n\n

A<\/p>\n\n\n\n

cannot be less than one.<\/p>\n\n\n\n

B<\/p>\n\n\n\n

cannot be one.<\/p>\n\n\n\n

C<\/p>\n\n\n\n

cannot be less than one and cannot be one.<\/p>\n\n\n\n

D<\/p>\n\n\n\n

None of the other answers is correct.<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

The correct answer is D) None of the other answers is correct.<\/strong><\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

Let’s denote two events as ( A ) and ( B ). The probability of the union of two events, ( P(A \\cup B) ), is the probability that at least one of the events ( A ) or ( B ) occurs. It is given by the formula:<\/p>\n\n\n\n

[
P(A \\cup B) = P(A) + P(B) – P(A \\cap B)
]<\/p>\n\n\n\n

This formula accounts for the fact that the probability of both events happening simultaneously (i.e., the intersection ( A \\cap B )) is subtracted to avoid double-counting.<\/p>\n\n\n\n

Now, let’s analyze each of the given options:<\/p>\n\n\n\n

    \n
  1. A) Cannot be less than one.<\/strong><\/li>\n<\/ol>\n\n\n\n
      \n
    • This is incorrect. While the probability of an event is always between 0 and 1, the union of two events can never exceed a probability of 1. For example, if ( P(A) = 0.8 ) and ( P(B) = 0.7 ), the probability of ( P(A \\cup B) ) can be at most 1, but it is typically less than 1 unless the events are perfectly correlated.<\/li>\n<\/ul>\n\n\n\n
        \n
      1. B) Cannot be one.<\/strong><\/li>\n<\/ol>\n\n\n\n
          \n
        • This is also incorrect. It is possible for the union of two events to have a probability of 1. For instance, if ( A ) and ( B ) cover the entire sample space (i.e., one or both will always occur), the union ( A \\cup B ) will have a probability of 1.<\/li>\n<\/ul>\n\n\n\n
            \n
          1. C) Cannot be less than one and cannot be one.<\/strong><\/li>\n<\/ol>\n\n\n\n
              \n
            • This is incorrect as well. The union of two events can indeed be less than 1 or equal to 1, depending on the relationship between the events and their probabilities.<\/li>\n<\/ul>\n\n\n\n
                \n
              1. D) None of the other answers is correct.<\/strong><\/li>\n<\/ol>\n\n\n\n
                  \n
                • This is the correct answer. The probability of the union of two events can range from 0 to 1, inclusive, depending on the overlap between the events. There is no restriction that prevents it from being less than 1 or equal to 1.<\/li>\n<\/ul>\n\n\n\n

                  Summary:<\/h3>\n\n\n\n

                  The probability of the union of two events ( A \\cup B ) can indeed be any value between 0 and 1, depending on the individual probabilities of ( A ) and ( B ), and their intersection. Therefore, none of the other provided options is entirely correct.<\/p>\n","protected":false},"excerpt":{"rendered":"

                  The probability of the union of two events with nonzero probabilities A cannot be less than one. B cannot be one. C cannot be less than one and cannot be one. D None of the other answers is correct. The Correct Answer and Explanation is : The correct answer is D) None of the other […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","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-5)","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-5)","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-5)","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":[25],"tags":[],"class_list":["post-188798","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188798","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=188798"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188798\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=188798"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=188798"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=188798"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}