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問１~~~四人がじゃんけんをする。４回目に初めて一人の\\[0.2cm]
勝者が決る確率を求めよ。\\[0.5cm]
解）\\[0.2cm]
一回目、二回目、三回目、四回目にじゃんけんをした後、それぞれの\\[0.2cm]
残りの人数が次のようになった場合が、４回目に初めて一人の勝者\\[0.2cm]
が決る場合である。\\[0.2cm]
~~~~~case 1)~~~4,4,4,1~~~\\[0.2cm]
~~~~~case 2)~~~4,4,3,1~~~\\[0.2cm]
~~~~~case 3)~~~4,4,2,1~~~\\[0.2cm]
~~~~~case 4)~~~4,3,3,1~~~\\[0.2cm]
~~~~~case 5)~~~4,3,2,1~~~\\[0.2cm]
~~~~~case 6)~~~3,3,3,1~~~\\[0.2cm]
~~~~~case 7)~~~3,3,2,1~~~\\[0.2cm]
~~~~~case 8)~~~3,2,2,1~~~\\[0.2cm]
~~~~~case 9)~~~2,2,2,1~~~\\[0.2cm]
\\
ここで、４人が一回のじゃんけんで、１人残る、２人残る、３人残る、\\[0.2cm]
４人残る確率を求める。\\[0.2cm]
(g+c+p)^4=g^4+c^4+p^4+4g^3c+4g^3p\\[0.2cm]
~~~~~~~~~~~~~~~~~+4c^3g+4c^3p\\[0.2cm]
~~~~~~~~~~~~~~~~~+4p^3g+4p^3c\\[0.2cm]
~~~~~~~~~~~+6g^2c^2+6g^2p^2+6c^2p^2\\[0.2cm]
~~~~~~~~~~~+12g^2cp+12c^2gp+12p^2gc\\[0.2cm]
g^4、c^4、p^4、12g^2cp、12c^2gp、12p^2gc~~があいこ。
即ち、４人残る確率~~(4-4)~~は\\[0.2cm]
\displaystyle\genfrac{}{}{0.5pt}{}{39}{81}
=\genfrac{}{}{0.5pt}{}{13}{27}\\[0.2cm]
4g^3c、4c^3p、4p^3g~~が三人勝ち。即ち、３人残る確率~~(4-3)~~は\\[0.2cm]
\genfrac{}{}{0.5pt}{}{12}{81}
=\genfrac{}{}{0.5pt}{}{4}{27}\\[0.2cm]
6g^2c^2、6g^2c^2、6c^2p^2~~が二人勝ち。即ち、２人残る確率~~(4-2)~~は\\[0.2cm]
\genfrac{}{}{0.5pt}{}{18}{81}
=\genfrac{}{}{0.5pt}{}{6}{27}\\[0.2cm]
4g^3p、4c^3g、4p^3c~~が一人勝ち。即ち、１人残る確率~~(4-1)~~は\\[0.2cm]
\genfrac{}{}{0.5pt}{}{12}{81}
=\genfrac{}{}{0.5pt}{}{4}{27}\\[0.2cm]
\\
また、３人が一回のじゃんけんで、１人残る、２人残る、３人残る、\\[0.2cm]
確率を求める。\\[0.2cm]
(g+c+p)^3=g^3+c^3+p^3+3g^2c+3g^2p\\[0.2cm]
~~~~~~~~~~~~~~~~~+3c^2g+3c^2p\\[0.2cm]
~~~~~~~~~~~~~~~~~+3p^2g+3p^2c\\[0.2cm]
~~~~~~~~~~~~~~~~~+6gcp\\[0.2cm]
g^3、c^3、p^3、6gcp~~があいこ。即ち、３人残る確率~~(3-3)~~は\\[0.2cm]
\displaystyle\genfrac{}{}{0.5pt}{}{9}{27}
=\genfrac{}{}{0.5pt}{}{1}{9}\\[0.2cm]
3g^2c、3c^2p、3p^2g~~が二人勝ち。即ち、２人残る確率~~(3-2)~~は\\[0.2cm]
\genfrac{}{}{0.5pt}{}{9}{27}
=\genfrac{}{}{0.5pt}{}{1}{9}\\[0.2cm]
3g^2p、3c^2g、3p^2c~~が一人勝ち。即ち、１人残る確率~~(3-1)~~は\\[0.2cm]
\genfrac{}{}{0.5pt}{}{9}{27}
=\genfrac{}{}{0.5pt}{}{1}{9}\\[0.2cm]
\\
また、２人が一回のじゃんけんで、１人残る、２人残る、\\[0.2cm]
確率を求める。\\[0.2cm]
(g+c+p)^2=g^2+c^2+p^2+2gc+2gp+2cp\\[0.2cm]
g^2、c^2、p^2、~~があいこ。即ち、２人残る確率~~(2-2)~~は\\[0.2cm]
\genfrac{}{}{0.5pt}{}{3}{9}
=\genfrac{}{}{0.5pt}{}{1}{3}\\[0.2cm]
2gc、2gp、2cp~~が一人勝ち。即ち、１人残る確率~~(2-1)~~は\\[0.2cm]
\genfrac{}{}{0.5pt}{}{6}{9}
=\genfrac{}{}{0.5pt}{}{2}{3}\\[0.2cm]
\\
さて、上の９case~のそれぞれの確率を求める。\\[0.2cm]
case1)~~(4-4)(4-4)(4-4)(4-1)~~で、\\[0.2cm]
~~~~~~\genfrac{}{}{0.5pt}{}{13}{27}\genfrac{}{}{0.5pt}{}{13}{27}
\genfrac{}{}{0.5pt}{}{13}{27}\genfrac{}{}{0.5pt}{}{4}{27}=
\genfrac{}{}{0.5pt}{}{8788}{531441}\\[0.2cm]
case2)~~(4-4)(4-4)(4-3)(3-1)~~で、\\[0.2cm]
~~~~~~\genfrac{}{}{0.5pt}{}{13}{27}\genfrac{}{}{0.5pt}{}{13}{27}
\genfrac{}{}{0.5pt}{}{4}{27}\genfrac{}{}{0.5pt}{}{9}{27}=
\genfrac{}{}{0.5pt}{}{6084}{531441}\\[0.2cm]
case3)~~(4-4)(4-4)(4-2)(2-1)~~で、\\[0.2cm]
~~~~~~\genfrac{}{}{0.5pt}{}{13}{27}\genfrac{}{}{0.5pt}{}{13}{27}
\genfrac{}{}{0.5pt}{}{9}{27}\genfrac{}{}{0.5pt}{}{6}{9}=
\genfrac{}{}{0.5pt}{}{18252}{531441}\\[0.2cm]
case4)~~(4-4)(4-3)(3-3)(3-1)~~で、\\[0.2cm]
~~~~~~\genfrac{}{}{0.5pt}{}{13}{27}\genfrac{}{}{0.5pt}{}{4}{27}
\genfrac{}{}{0.5pt}{}{9}{27}\genfrac{}{}{0.5pt}{}{9}{27}=
\genfrac{}{}{0.5pt}{}{4212}{531441}\\[0.2cm]
case5)~~(4-3)(3-3)(3-2)(2-1)~~で、\\[0.2cm]
~~~~~~\genfrac{}{}{0.5pt}{}{13}{27}\genfrac{}{}{0.5pt}{}{4}{27}
\genfrac{}{}{0.5pt}{}{9}{27}\genfrac{}{}{0.5pt}{}{6}{9}=
\genfrac{}{}{0.5pt}{}{8424}{531441}\\[0.2cm]
case6)~~(4-3)(3-3)(3-3)(3-1)~~で、\\[0.2cm]
~~~~~~\genfrac{}{}{0.5pt}{}{4}{27}\genfrac{}{}{0.5pt}{}{9}{27}
\genfrac{}{}{0.5pt}{}{9}{27}\genfrac{}{}{0.5pt}{}{9}{27}=
\genfrac{}{}{0.5pt}{}{2916}{531441}\\[0.2cm]
case7)~~(4-3)(3-3)(3-2)(2-1)~~で、\\[0.2cm]
~~~~~~\genfrac{}{}{0.5pt}{}{4}{27}\genfrac{}{}{0.5pt}{}{9}{27}
\genfrac{}{}{0.5pt}{}{9}{27}\genfrac{}{}{0.5pt}{}{6}{9}=
\genfrac{}{}{0.5pt}{}{5832}{531441}\\[0.2cm]
case8)~~(4-3)(3-2)(2-2)(2-1)~~で、\\[0.2cm]
~~~~~~\genfrac{}{}{0.5pt}{}{4}{27}\genfrac{}{}{0.5pt}{}{9}{27}
\genfrac{}{}{0.5pt}{}{3}{9}\genfrac{}{}{0.5pt}{}{6}{9}=
\genfrac{}{}{0.5pt}{}{5832}{531441}\\[0.2cm]
case9)~~(4-2)(2-2)(2-2)(2-1)~~で、\\[0.2cm]
~~~~~~\genfrac{}{}{0.5pt}{}{6}{27}\genfrac{}{}{0.5pt}{}{3}{9}
\genfrac{}{}{0.5pt}{}{3}{9}\genfrac{}{}{0.5pt}{}{6}{9}=
\genfrac{}{}{0.5pt}{}{8748}{531441}\\[0.2cm]
これらを加えて、
~~~~~~~\genfrac{}{}{0.5pt}{}{69097}{531441}~~~~~（答）
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