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标签：body &# color token html 测试 toc

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11.投球游戏规定如下：有①②③④四个区域，投进①或②一次，或者投进③或④两次则胜利。假如某同学投球一定能投进设投进①或②的概率为<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>p，投进③的概率为<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi>p</mi></mrow><annotation encoding="application/x-tex">4p</annotation></semantics></math>4p，记两次投球后恰好获胜的概率为<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">P_1</annotation></semantics></math>P1,四次投球后恰好胜利的概率为<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">P_2</annotation></semantics></math>P2，下列说法正确的有： 
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mrow><mtext mathvariant="bold">答案：</mtext><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">C</mi></mrow></mi></mrow><annotation encoding="application/x-tex">\bm{答案：AC}</annotation></semantics></math>答案：AC 
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi mathvariant="normal">.</mi><mn>0</mn><mo><</mo><mi>p</mi><mo><</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">A.0<p<\dfrac{1}{6}</annotation></semantics></math>A.0<p<61 
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi mathvariant="normal">.</mi><msub><mi>P</mi><mn>1</mn></msub><mo>=</mo><mn>16</mn><msup><mi>p</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">B.P_1 = 16p^2</annotation></semantics></math>B.P1=16p2 
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi mathvariant="normal">.</mi><msub><mi>P</mi><mn>2</mn></msub><mo>=</mo><mn>12</mn><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>36</mn><msup><mi>p</mi><mn>3</mn></msup><mo>−</mo><mn>12</mn><msup><mi>p</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C.P_2 = 12(p+36p^3-12p^2)</annotation></semantics></math>C.P2=12(p+36p3−12p2) 
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi mathvariant="normal">.</mi><mtext>若</mtext><msub><mi>P</mi><mn>1</mn></msub><mo><</mo><msub><mi>P</mi><mn>2</mn></msub><mtext>，则</mtext><mi>p</mi><mtext>的取值范围为</mtext><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D.若P_1<P_2，则p的取值范围为(0,\dfrac{1}{8})</annotation></semantics></math>D.若P1<P2，则p的取值范围为(0,81)
投进④的概率为<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mn>6</mn><mi>p</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1-6p\in(0,1)</annotation></semantics></math>1−6p∈(0,1),得<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p\in (0,\dfrac{1}{6})</annotation></semantics></math>p∈(0,61) 
若两次投球后恰好获胜，则第一次只能投进③或者④(否则投进①或②就一次获胜了) 
则有以下情况： 
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>③①，③②，③③</mtext></mrow><annotation encoding="application/x-tex">③①，③②，③③</annotation></semantics></math>③①，③②，③③ 
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>④①，④②</mtext></mrow><annotation encoding="application/x-tex">④①，④②</annotation></semantics></math>④①，④② 
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mn>1</mn></msub><mo>=</mo><mn>4</mn><mi>p</mi><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>p</mi><mo>+</mo><mn>4</mn><mi>p</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>6</mn><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mn>12</mn><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>p</mi></mrow><annotation encoding="application/x-tex">P_1 = 4p(p+p+4p)+(1-6p)(p+p) = 12p^2+2p</annotation></semantics></math>P1=4p(p+p+4p)+(1−6p)(p+p)=12p2+2p 
若四次投球恰好获胜，先考虑前三次，显然前三次不能投进①或者②，不然直接获胜，而前三次不能都投进④，也不能有两次投进③，所以前三次必然有一次投进③有两次投进④ 
前三次有一次投进③有两次投进④，无论第四次投进哪个区域，都会获胜，所以<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mn>2</mn></msub><mo>=</mo><msubsup><mi>C</mi><mn>3</mn><mn>1</mn></msubsup><mn>4</mn><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>6</mn><mi>p</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mn>12</mn><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mn>12</mn><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mn>36</mn><msup><mi>p</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_2 = C_3^14p(1-6p)^2 = 12(p-12p^2+36p^3)</annotation></semantics></math>P2=C314p(1−6p)2=12(p−12p2+36p3)
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>令</mtext><msub><mi>P</mi><mn>2</mn></msub><mo>></mo><msub><mi>P</mi><mn>1</mn></msub><mo separator="true">,</mo><mtext>得</mtext><mn>216</mn><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><mn>78</mn><mi>p</mi><mo>+</mo><mn>5</mn><mo>></mo><mn>0</mn><mo separator="true">,</mo><mtext>对称轴为</mtext><mi>p</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>13</mn><mn>72</mn></mfrac></mstyle><mo>></mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">令P_2>P_1,得216p^2-78p+5>0,对称轴为 p = \dfrac{13}{72}>\dfrac{1}{8}</annotation></semantics></math>令P2>P1,得216p2−78p+5>0,对称轴为p=7213>81 
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>令</mtext><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mn>216</mn><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><mn>78</mn><mi>p</mi><mo>+</mo><mn>5</mn><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mtext>在</mtext><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle><mo stretchy="false">)</mo><mtext>上单调递减，</mtext><mi>f</mi><mo stretchy="false">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle><mo stretchy="false">)</mo><mo><</mo><mn>0</mn><mo separator="true">,</mo><mtext>则</mtext><mi>p</mi><mtext>的取值范围不可能是</mtext><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">令f(p) = 216p^2-78p+5,f(p)在(0,\dfrac{1}{8})上单调递减，f(\dfrac{1}{8})<0,则p的取值范围不可能是(0,\dfrac{1}{8}).</annotation></semantics></math>令f(p)=216p2−78p+5,f(p)在(0,81)上单调递减，f(81)<0,则p的取值范围不可能是(0,81).

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