回復 2# jasonmv6124 的帖子
第 4 大題第 (2) 小題
設擲出一、二、三、四、五、六點的機率，分別是\({{p}_{1}},{{p}_{2}},{{p}_{3}},{{p}_{4}},{{p}_{5}},{{p}_{6}}\)
\(\begin{align}
& {{p}_{1}}+{{p}_{2}}+{{p}_{3}}+{{p}_{4}}+{{p}_{5}}+{{p}_{6}}=1 \\
& T=2\left( {{p}_{1}}+{{p}_{3}}+{{p}_{5}} \right)\left( {{p}_{2}}+{{p}_{4}}+{{p}_{6}} \right) \\
& \frac{T}{2}=\left( {{p}_{1}}+{{p}_{3}}+{{p}_{5}} \right)\left( {{p}_{2}}+{{p}_{4}}+{{p}_{6}} \right)\le {{\left[ \frac{\left( {{p}_{1}}+{{p}_{3}}+{{p}_{5}} \right)+\left( {{p}_{2}}+{{p}_{4}}+{{p}_{6}} \right)}{2} \right]}^{2}}=\frac{1}{4} \\
& T\le \frac{1}{2} \\
\end{align}\)
\(\begin{align}
& S={{p}_{1}}^{2}+{{p}_{2}}^{2}+{{p}_{3}}^{2}+{{p}_{4}}^{2}+{{p}_{5}}^{2}+{{p}_{6}}^{2} \\
& W={{p}_{1}}{{p}_{3}}+{{p}_{1}}{{p}_{5}}+{{p}_{2}}{{p}_{4}}+{{p}_{2}}{{p}_{6}}+{{p}_{3}}{{p}_{1}}+{{p}_{3}}{{p}_{5}}+{{p}_{4}}{{p}_{2}}+{{p}_{4}}{{p}_{6}}+{{p}_{5}}{{p}_{1}}+{{p}_{5}}{{p}_{3}}+{{p}_{6}}{{p}_{2}}+{{p}_{6}}{{p}_{4}} \\
& =\left( {{p}_{1}}{{p}_{3}}+{{p}_{2}}{{p}_{4}}+{{p}_{3}}{{p}_{5}}+{{p}_{4}}{{p}_{6}}+{{p}_{5}}{{p}_{1}}+{{p}_{6}}{{p}_{2}} \right)+\left( {{p}_{1}}{{p}_{5}}+{{p}_{2}}{{p}_{6}}+{{p}_{3}}{{p}_{1}}+{{p}_{4}}{{p}_{2}}+{{p}_{5}}{{p}_{3}}+{{p}_{6}}{{p}_{4}} \right) \\
& T+S+W=1 \\
& W\le S+S=2S \\
& 3S+T=2S+S+T\ge W+S+T=1 \\
& T\ge 1-3S \\
& \\
& \frac{1}{2}\ge T\ge 1-3S \\
\end{align}\)
[ 本帖最後由 thepiano 於 2019-4-22 08:31 編輯 ]
