{"id":157,"date":"2019-12-31T17:45:00","date_gmt":"2019-12-31T17:45:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=157"},"modified":"2020-10-16T20:19:03","modified_gmt":"2020-10-16T20:19:03","slug":"mathcounts-exercises-7","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=157","title":{"rendered":"MATHCOUNTS Exercises &#8211; 7"},"content":{"rendered":"\n<p>John has two coins in his pocket. One of them is a fair coin, the other is crooked that will land on head 2 out of 3 throws. John randomly selects one of the coins from his pocket, and flips it 2 times. If the coin lands on head both times, what is the probability that the coin is the fair coin? <a onclick='toggle_visibility(\"solution-mathcounts-2-coins\");'>Click here for the solution.<\/a><\/p>\n<div id='solution-mathcounts-2-coins' style='display:none'>\n<p><strong>Solution<\/strong>\nLet \\(P_{fair}\\) and \\(P_{crooked}\\) denote the probability of the fair coin and the crooked coin landing on head. We have $$P_{fair} = \\frac{1}{2}, \\ \\ \\ P_{crooked} = \\frac{2}{3}$$ The probability of selecting the fair coin or the crooked coin randomly is \\(\\dfrac{1}{2}\\). Therefore the probability of both flips landing on head is $$\\frac{1}{2}\\cdot (P_{fair})^2+ \\frac{1}{2}\\cdot (P_{crooked})^2 = \\frac{1}{2}\\cdot(\\frac{1}{2})^2+\\frac{1}{2}\\cdot(\\frac{2}{3})^2=\\frac{1}{8}+\\frac{2}{9}=\\frac{25}{72}$$ Therefore, the probability of the coin is the fair coin is $$ \\frac{\\dfrac{1}{8}}{\\dfrac{25}{72}}=\\frac{9}{25}$$ \\(\\blacksquare\\)\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>John has two coins in his pocket. One of them is a fair coin, the other is crooked that will land on head 2 out of 3 throws. John randomly selects one of the coins from his pocket, and flips &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=157\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[11,8],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/157"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=157"}],"version-history":[{"count":2,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/157\/revisions"}],"predecessor-version":[{"id":566,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/157\/revisions\/566"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}