【数学A】例題2.2.7：反復試行の確率（平面上の点の移動）（One More）★★★

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% 例題A2.2.7：反復試行の確率（平面上の点の移動）（One More）★★★
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右の図のような格子状の道路がある．A地点からB地点まで最短経路で行くとき，次の問いに答えよ．ただし，各交差点において，東，北のいずれの進路も進む確率は，ともに$\frac{1}{2}$であり，一方にしか進めないときは確率1でその方向に進むものとする．

(1) C地点を通る確率を求めよ．

(2) D地点を通る確率を求めよ．

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(1) A地点からC地点へは，東へ2区画，北へ3区画進む必要があるから，求める確率は，

\[
{}_5\mathrm{C}_2(\frac{1}{2})^2(\frac{1}{2})^3=\frac{5}{16}
\]

(2) 右の図のように，地点P，Q，$\mathrm{P}^\prime$，$\mathrm{Q}^\prime$をとる．Dを通る道順は3つの場合があり，その確率はそれぞれ次のようになる．

(i) $\mathrm{A}$から$\mathrm{P}^\prime,\mathrm{P}$を通りDに行くとき，$\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times 1\times 1=\frac{1}{16}$

(ii) $\mathrm{A}$から$\mathrm{Q}^\prime,\mathrm{Q}$を通りDに行くとき，${}_4\mathrm{C}_1(\frac{1}{2})^1(\frac{1}{2})^3\times\frac{1}{2}\times 1=\frac{1}{8}$

(iii) $\mathrm{A}$から$\mathrm{C}$を通りDに行くとき，(1)の結果を用いて，$\frac{5}{16}\times\frac{1}{2}=\frac{5}{32}$

よって，(i)〜(iii)は互いに排反であるから，求める確率は，

\[
\frac{1}{16}+\frac{1}{8}+\frac{5}{32}=\frac{11}{32}
\]

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% 問題A2.2.7：反復試行の確率（平面上の点の移動）（One More）★★★
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右の図のような格子状の道路がある．A地点からB地点まで最短経路で行くとき，C地点を通る確率を求めよ．ただし，各交差点において，東，北のいずれの進路も進む確率は，ともに$\frac{1}{2}$であり，一方にしか進めないときは確率1でその方向に進むものとする．

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右の図のように，地点P，Qをとる．Cを通る道順は2つの場合があり，その確率はそれぞれ次のようになる．

(i) $\mathrm{A}$からPを通りCに行くとき，$\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times 1=\frac{1}{8}$

(ii) $\mathrm{A}$から$\mathrm{Q}$を通りCに行くとき

A地点からQ地点へは，東へ1区画，北へ2区画進む必要があるから，その確率は，

\[
{}_3\mathrm{C}_1(\frac{1}{2})(\frac{1}{2})^2=\frac{3}{8}
\]

したがって，$\mathrm{A}$から$\mathrm{Q}$を通りCに行く確率は，
\[
\frac{3}{8}\times\frac{1}{2}=\frac{3}{16}
\]

よって，(i)，(ii)は互いに排反であるから，求める確率は，

\[
\frac{1}{8}+\frac{3}{16}=\frac{5}{16}
\]

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% 例題A2.2.7の側注：反復試行の確率（平面上の点の移動）（One More）★★★
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% 側注1
$\uparrow\uparrow\uparrow\uparrow\rightarrow\rightarrow$と進む．

% 側注2
$\ldots,\uparrow\rightarrow$と進む．

% 側注3
$\ldots,\uparrow$と進む．

% 側注4
C地点を通った後のことは考えなくてよい．

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% 側注1
P地点を通るか，Q地点を通るかで場合分けをする．

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% 例題A2.2.7の考え方・補足など：反復試行の確率（平面上の点の移動）（One More）★★★
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(1) 求める確率を，$\frac{(\mathrm{A}\text{から}\mathrm{C}\text{を通り}\mathrm{B}\text{に行く最短経路の総数})}{(\mathrm{A}\text{から}\mathrm{B}\text{に行く最短経路の総数})}$と考え，$\frac{{}_5\mathrm{C}_3\times{}_5\mathrm{C}_1}{{}_{10}\mathrm{C}_4}$などとするのは，道順によって確率が異なる（同様に確からしくない）ので誤りである．右へ1区画進むことを$\rightarrow$，上へ1区画進むことを$\uparrow$と表し，例えば次のような道順を考えると，確率が異なることがわかるので注意すること．

(i) $\rightarrow\uparrow\uparrow\rightarrow\uparrow\rightarrow\rightarrow\rightarrow\rightarrow\uparrow$のとき，その確率は，

\[
\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times 1=\frac{1}{512}
\]

(ii) $\uparrow\uparrow\uparrow\rightarrow\rightarrow\uparrow\rightarrow\rightarrow\rightarrow\rightarrow$のとき，その確率は，

\[
\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times 1\times 1\times 1\times 1=\frac{1}{64}
\]

(2) Dを通る道順を，通る点で場合分けして確率を求める．