{"id":90018,"date":"2025-06-01T10:19:13","date_gmt":"2025-06-01T10:19:13","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90018"},"modified":"2025-06-01T10:19:13","modified_gmt":"2025-06-01T10:19:13","slug":"there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/","title":{"rendered":"There are two boxes. Box I contains one white card and two black cards"},"content":{"rendered":"

There are two boxes. Box I contains one white card and two black cards and Box II contains one white card and a black card. Two persons P and Q play a game. P picks a card randomly from Box I. If P finds the white card, P wins and the game stops. If P finds the black card, Q draws a card randomly from Box II. If Q finds the white card, Q wins. The game stops whether Q draws the white card or the black card. Which one of the following is correct?<\/p>\n

[amp_mcq option1=”If P loses, Q wins” option2=”If Q loses, P wins” option3=”Both P and Q may win” option4=”Both P and Q may lose” correct=”option4″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2016<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct option is D. The game setup allows for a scenario where neither P nor Q wins.
\n<\/section>\n
\nThe game proceeds as follows:
\n1. P draws a card from Box I (1 White, 2 Black).
\n * If P draws White (probability 1\/3), P wins and the game stops.
\n * If P draws Black (probability 2\/3), P loses this step, and Q draws from Box II.
\n2. If P drew Black, Q draws a card from Box II (1 White, 1 Black).
\n * If Q draws White (probability 1\/2), Q wins. The game stops.
\n * If Q draws Black (probability 1\/2), Q loses. The game stops.<\/p>\n

Let’s analyze the outcomes:
\n* Outcome 1: P draws White (Prob=1\/3). P wins.
\n* Outcome 2: P draws Black (Prob=2\/3) AND Q draws White (Prob=1\/2). Q wins. Probability = (2\/3)*(1\/2) = 1\/3.
\n* Outcome 3: P draws Black (Prob=2\/3) AND Q draws Black (Prob=1\/2). Neither P nor Q wins. Probability = (2\/3)*(1\/2) = 1\/3.<\/p>\n

Now let’s evaluate the options:
\nA) If P loses, Q wins: P loses if P draws Black. If P draws Black, Q draws from Box II. Q wins *only if* Q draws White. Q does *not* win if Q draws Black. So this statement is not always correct.
\nB) If Q loses, P wins: Q only plays if P loses (draws Black). If Q loses (draws Black), it means P already lost the first draw. P’s winning condition is drawing White in the *first* step. If Q gets to play and then loses, P cannot win *in that game instance*. So this statement is incorrect.
\nC) Both P and Q may win: In a single game instance, either P wins (game stops), or P loses and Q plays. If Q plays, either Q wins or neither wins. P and Q cannot both win in the same game. So this statement is incorrect.
\nD) Both P and Q may lose: This happens in Outcome 3, where P draws Black and Q draws Black. In this scenario, P did not win (as P drew Black) and Q did not win (as Q drew Black). This is a possible outcome with probability 1\/3. So this statement is correct.
\n<\/section>\n

\nThe total probability of winning for P is 1\/3. The total probability of winning for Q is 1\/3. The probability that neither wins is 1\/3. The sum of probabilities is 1\/3 + 1\/3 + 1\/3 = 1.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

There are two boxes. Box I contains one white card and two black cards and Box II contains one white card and a black card. Two persons P and Q play a game. P picks a card randomly from Box I. If P finds the white card, P wins and the game stops. If P … <\/p>\n

Detailed SolutionThere are two boxes. Box I contains one white card and two black cards<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1098,1102],"class_list":["post-90018","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1098","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nThere are two boxes. Box I contains one white card and two black cards<\/title>\n<meta name=\"description\" content=\"The correct option is D. The game setup allows for a scenario where neither P nor Q wins. The game proceeds as follows: 1. P draws a card from Box I (1 White, 2 Black). * If P draws White (probability 1\/3), P wins and the game stops. * If P draws Black (probability 2\/3), P loses this step, and Q draws from Box II. 2. If P drew Black, Q draws a card from Box II (1 White, 1 Black). * If Q draws White (probability 1\/2), Q wins. The game stops. * If Q draws Black (probability 1\/2), Q loses. The game stops. Let's analyze the outcomes: * Outcome 1: P draws White (Prob=1\/3). P wins. * Outcome 2: P draws Black (Prob=2\/3) AND Q draws White (Prob=1\/2). Q wins. Probability = (2\/3)*(1\/2) = 1\/3. * Outcome 3: P draws Black (Prob=2\/3) AND Q draws Black (Prob=1\/2). Neither P nor Q wins. Probability = (2\/3)*(1\/2) = 1\/3. Now let's evaluate the options: A) If P loses, Q wins: P loses if P draws Black. If P draws Black, Q draws from Box II. Q wins *only if* Q draws White. Q does *not* win if Q draws Black. So this statement is not always correct. B) If Q loses, P wins: Q only plays if P loses (draws Black). If Q loses (draws Black), it means P already lost the first draw. P's winning condition is drawing White in the *first* step. If Q gets to play and then loses, P cannot win *in that game instance*. So this statement is incorrect. C) Both P and Q may win: In a single game instance, either P wins (game stops), or P loses and Q plays. If Q plays, either Q wins or neither wins. P and Q cannot both win in the same game. So this statement is incorrect. D) Both P and Q may lose: This happens in Outcome 3, where P draws Black and Q draws Black. In this scenario, P did not win (as P drew Black) and Q did not win (as Q drew Black). This is a possible outcome with probability 1\/3. So this statement is correct.\" \/>\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\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"There are two boxes. Box I contains one white card and two black cards\" \/>\n<meta property=\"og:description\" content=\"The correct option is D. The game setup allows for a scenario where neither P nor Q wins. The game proceeds as follows: 1. P draws a card from Box I (1 White, 2 Black). * If P draws White (probability 1\/3), P wins and the game stops. * If P draws Black (probability 2\/3), P loses this step, and Q draws from Box II. 2. If P drew Black, Q draws a card from Box II (1 White, 1 Black). * If Q draws White (probability 1\/2), Q wins. The game stops. * If Q draws Black (probability 1\/2), Q loses. The game stops. Let's analyze the outcomes: * Outcome 1: P draws White (Prob=1\/3). P wins. * Outcome 2: P draws Black (Prob=2\/3) AND Q draws White (Prob=1\/2). Q wins. Probability = (2\/3)*(1\/2) = 1\/3. * Outcome 3: P draws Black (Prob=2\/3) AND Q draws Black (Prob=1\/2). Neither P nor Q wins. Probability = (2\/3)*(1\/2) = 1\/3. Now let's evaluate the options: A) If P loses, Q wins: P loses if P draws Black. If P draws Black, Q draws from Box II. Q wins *only if* Q draws White. Q does *not* win if Q draws Black. So this statement is not always correct. B) If Q loses, P wins: Q only plays if P loses (draws Black). If Q loses (draws Black), it means P already lost the first draw. P's winning condition is drawing White in the *first* step. If Q gets to play and then loses, P cannot win *in that game instance*. So this statement is incorrect. C) Both P and Q may win: In a single game instance, either P wins (game stops), or P loses and Q plays. If Q plays, either Q wins or neither wins. P and Q cannot both win in the same game. So this statement is incorrect. D) Both P and Q may lose: This happens in Outcome 3, where P draws Black and Q draws Black. In this scenario, P did not win (as P drew Black) and Q did not win (as Q drew Black). This is a possible outcome with probability 1\/3. So this statement is correct.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:19:13+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":"There are two boxes. Box I contains one white card and two black cards","description":"The correct option is D. The game setup allows for a scenario where neither P nor Q wins. The game proceeds as follows: 1. P draws a card from Box I (1 White, 2 Black). * If P draws White (probability 1\/3), P wins and the game stops. * If P draws Black (probability 2\/3), P loses this step, and Q draws from Box II. 2. If P drew Black, Q draws a card from Box II (1 White, 1 Black). * If Q draws White (probability 1\/2), Q wins. The game stops. * If Q draws Black (probability 1\/2), Q loses. The game stops. Let's analyze the outcomes: * Outcome 1: P draws White (Prob=1\/3). P wins. * Outcome 2: P draws Black (Prob=2\/3) AND Q draws White (Prob=1\/2). Q wins. Probability = (2\/3)*(1\/2) = 1\/3. * Outcome 3: P draws Black (Prob=2\/3) AND Q draws Black (Prob=1\/2). Neither P nor Q wins. Probability = (2\/3)*(1\/2) = 1\/3. Now let's evaluate the options: A) If P loses, Q wins: P loses if P draws Black. If P draws Black, Q draws from Box II. Q wins *only if* Q draws White. Q does *not* win if Q draws Black. So this statement is not always correct. B) If Q loses, P wins: Q only plays if P loses (draws Black). If Q loses (draws Black), it means P already lost the first draw. P's winning condition is drawing White in the *first* step. If Q gets to play and then loses, P cannot win *in that game instance*. So this statement is incorrect. C) Both P and Q may win: In a single game instance, either P wins (game stops), or P loses and Q plays. If Q plays, either Q wins or neither wins. P and Q cannot both win in the same game. So this statement is incorrect. D) Both P and Q may lose: This happens in Outcome 3, where P draws Black and Q draws Black. In this scenario, P did not win (as P drew Black) and Q did not win (as Q drew Black). This is a possible outcome with probability 1\/3. So this statement is correct.","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\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/","og_locale":"en_US","og_type":"article","og_title":"There are two boxes. Box I contains one white card and two black cards","og_description":"The correct option is D. The game setup allows for a scenario where neither P nor Q wins. The game proceeds as follows: 1. P draws a card from Box I (1 White, 2 Black). * If P draws White (probability 1\/3), P wins and the game stops. * If P draws Black (probability 2\/3), P loses this step, and Q draws from Box II. 2. If P drew Black, Q draws a card from Box II (1 White, 1 Black). * If Q draws White (probability 1\/2), Q wins. The game stops. * If Q draws Black (probability 1\/2), Q loses. The game stops. Let's analyze the outcomes: * Outcome 1: P draws White (Prob=1\/3). P wins. * Outcome 2: P draws Black (Prob=2\/3) AND Q draws White (Prob=1\/2). Q wins. Probability = (2\/3)*(1\/2) = 1\/3. * Outcome 3: P draws Black (Prob=2\/3) AND Q draws Black (Prob=1\/2). Neither P nor Q wins. Probability = (2\/3)*(1\/2) = 1\/3. Now let's evaluate the options: A) If P loses, Q wins: P loses if P draws Black. If P draws Black, Q draws from Box II. Q wins *only if* Q draws White. Q does *not* win if Q draws Black. So this statement is not always correct. B) If Q loses, P wins: Q only plays if P loses (draws Black). If Q loses (draws Black), it means P already lost the first draw. P's winning condition is drawing White in the *first* step. If Q gets to play and then loses, P cannot win *in that game instance*. So this statement is incorrect. C) Both P and Q may win: In a single game instance, either P wins (game stops), or P loses and Q plays. If Q plays, either Q wins or neither wins. P and Q cannot both win in the same game. So this statement is incorrect. D) Both P and Q may lose: This happens in Outcome 3, where P draws Black and Q draws Black. In this scenario, P did not win (as P drew Black) and Q did not win (as Q drew Black). This is a possible outcome with probability 1\/3. So this statement is correct.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:19:13+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\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/","url":"https:\/\/exam.pscnotes.com\/mcq\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/","name":"There are two boxes. Box I contains one white card and two black cards","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:19:13+00:00","dateModified":"2025-06-01T10:19:13+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct option is D. The game setup allows for a scenario where neither P nor Q wins. The game proceeds as follows: 1. P draws a card from Box I (1 White, 2 Black). * If P draws White (probability 1\/3), P wins and the game stops. * If P draws Black (probability 2\/3), P loses this step, and Q draws from Box II. 2. If P drew Black, Q draws a card from Box II (1 White, 1 Black). * If Q draws White (probability 1\/2), Q wins. The game stops. * If Q draws Black (probability 1\/2), Q loses. The game stops. Let's analyze the outcomes: * Outcome 1: P draws White (Prob=1\/3). P wins. * Outcome 2: P draws Black (Prob=2\/3) AND Q draws White (Prob=1\/2). Q wins. Probability = (2\/3)*(1\/2) = 1\/3. * Outcome 3: P draws Black (Prob=2\/3) AND Q draws Black (Prob=1\/2). Neither P nor Q wins. Probability = (2\/3)*(1\/2) = 1\/3. Now let's evaluate the options: A) If P loses, Q wins: P loses if P draws Black. If P draws Black, Q draws from Box II. Q wins *only if* Q draws White. Q does *not* win if Q draws Black. So this statement is not always correct. B) If Q loses, P wins: Q only plays if P loses (draws Black). If Q loses (draws Black), it means P already lost the first draw. P's winning condition is drawing White in the *first* step. If Q gets to play and then loses, P cannot win *in that game instance*. So this statement is incorrect. C) Both P and Q may win: In a single game instance, either P wins (game stops), or P loses and Q plays. If Q plays, either Q wins or neither wins. P and Q cannot both win in the same game. So this statement is incorrect. D) Both P and Q may lose: This happens in Outcome 3, where P draws Black and Q draws Black. In this scenario, P did not win (as P drew Black) and Q did not win (as Q drew Black). This is a possible outcome with probability 1\/3. So this statement is correct.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/there-are-two-boxes-box-i-contains-one-white-card-and-two-black-cards\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"There are two boxes. Box I contains one white card and two black cards"}]},{"@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\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?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\/90018","targetHints":{"allow":["GET"]}}],"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=90018"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90018\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=90018"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=90018"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=90018"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}