{"id":2828,"date":"2022-08-29T23:42:00","date_gmt":"2022-08-30T03:42:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2828"},"modified":"2024-10-25T11:05:20","modified_gmt":"2024-10-25T15:05:20","slug":"probability-challenge-2022-08-29","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2828","title":{"rendered":"Probability Challenge 2022\/08\/29"},"content":{"rendered":"\n

A frog can randomly jump exactly 1 yard away in any random direction constantly. (1)<\/strong> What is the probability that the frog is within 1 yard away from its starting point after 2 jumps? (2)<\/strong> What is the probability that the frog is within 1 yard away from its starting point after 3 jumps? Click here<\/a> for the solution.\n<\/p>\n\n\n\n

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Solution (1): <\/strong>Assume the frog starts at point $A$, and its first jump lands at point $B$:<\/p>\n\n\n\n

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In order to land the second jump with 1 yard from point $A$, the frog must be land at any point on arc $\\overset{\\Large\\frown}{CAD}$. Since length of arc $\\overset{\\Large\\frown}{CAD}$ is $\\dfrac{1}{3}$ of the circumference of the circle centered at point $B$, the probability for the second jump lands within 1 yard from point $A$ is $\\boxed{\\dfrac{1}{3}}$.<\/p>\n\n\n\n

Solution (2):<\/strong> Assume the frog starts at point $A$ with its coordinate as $(1,0)$, and its first jump lands at point $B$ with its coordinate as $(0,0)$, and the second jump lands at point $C$, and the third jump lands at point $D$.<\/p>\n\n\n\n

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Assume the directional angle along the $x$-axis of the second jump is $\\alpha\\ (0\\le\\alpha<2\\pi)$, and that of the third jump is $\\beta\\ (0\\le\\beta<2\\pi)$ . Then, the coordinate of point $D$ would be $(\\cos\\alpha+\\cos\\beta,\\sin\\alpha + \\sin\\beta)$. Because the distance between $A$ and $D$ must be within 1 yard, therefore $$(\\cos\\alpha+\\cos\\beta-1)^2+(\\sin\\alpha+\\sin\\beta)^2\\le 1 \\tag{1}$$ Simplify the above equation, we have $$(\\cos\\alpha-1)(\\cos\\beta-1)+\\sin\\alpha\\cdot\\sin\\beta\\le 0 \\tag{2}$$ The above inequality can be transformed into the following: $$\\sin\\dfrac{\\alpha}{2}\\cdot\\sin\\dfrac{\\beta}{2}\\cdot\\cos\\dfrac{\\alpha-\\beta}{2}\\le 0 \\tag{3}$$ Because $0<=\\alpha<2\\pi$, and $0<=\\beta<2\\pi$, we have $\\sin\\dfrac{\\alpha}{2}\\ge 0$, $\\sin\\dfrac{\\beta}{2}\\ge 0$. Therefore $(3)$ can be transformed into the following$$\\cos\\dfrac{\\alpha-\\beta}{2}\\le 0 \\tag{4}$$ Solve the above inequality, we have $$0\\le\\alpha\\lt \\pi,\\ \\alpha+\\pi\\le\\beta\\le \\pi$$ $$ \\pi\\le\\alpha\\lt 2\\pi,\\ 0\\le\\beta\\le\\alpha-\\pi$$<\/p>\n\n\n\n

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As illustrated in the above diagram, the area of the red region, depicted the solutions of $\\alpha$ and $\\beta$ is $\\dfrac{1}{4}$ of the whole square for all possible $\\alpha$ and $\\beta$ values. Therefore, the probability of the third jump lands within 1 yard of the start point is $\\boxed{\\dfrac{1}{4}}$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

A frog can randomly jump exactly 1 yard away in any random direction constantly. (1) What is the probability that the frog is within 1 yard away from its starting point after 2 jumps? (2) What is the probability that … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13,8,15],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2828"}],"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=2828"}],"version-history":[{"count":37,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2828\/revisions"}],"predecessor-version":[{"id":4601,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2828\/revisions\/4601"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2828"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2828"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2828"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}