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(b-d)/(a-d)＜(b-c)/(a-c)＜b/a＜(b+c)/(a+c)＜(b+d)/(a+d)

\frac{b-d}{a-d} < \frac{b-c}{a-c} < \frac{b}{a} < \frac{b+c}{a+c} < \frac{b+d}{a+d}

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\frac{b}{a} < \frac{b+c}{a+c}
b(a+c) < (b+c)a
ab+bc < ab+ac
bc < ac
b < a

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\frac{b+c}{a+c} < \frac{b+d}{a+d}
(b+c)(a+d) < (b+d)(a+c)
ab+bd+ac+cd < ab+bc+ad+cd
ac+bd < ad+bc
b(d-c) < a(d-c)
0 < d-c ∵ b < a
c < d

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\frac{b-c}{a-c} < \frac{b}{a}
if 0 < a-c then
(b-c)a < b(a-c)
-ac < -bc
a > b OK
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0 < a-c
c < a

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if a-c < 0 then
(b-c)a > b(a-c)
-ac > -bc
a < b NG

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\frac{b-d}{a-d} < \frac{b-c}{a-c}
if 0 < a-d then
(b-d)(a-c) < (b-c)(a-d)
ab-ad-bc+cd < ab-ac-bd+cd
-ad-bc < -ac-bd
a(c-d) < b(c-d)
c-d < 0 ∵ b < a
c < d OK
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0 < a-d
d < a

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if a-d < 0 then
(b-d)(a-c) < (b-c)(a-d)
ab-ad-bc+cd > ab-ac-bd+cd
-ad-bc > -ac-bd
a(c-d) > b(c-d)
c-d > 0 ∵ b < a
c > d NG

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\frac{b-d}{a-d} < \frac{b-c}{a-c} < \frac{b}{a} < \frac{b+c}{a+c} < \frac{b+d}{a+d}
⇔　(b < a) and (c < d < a)
確率は\frac{1}{8}

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