将 1,2,3,4,5,6,7 任意分成两组，每组至少有一个数，求两组数字之和相等的概率

wintex

請問陸老師機率

luyuanhong

题  将 1,2,3,4,5,6,7 任意分成两组，每组至少有一个数，求两组数字之和相等的概率。

解  将 7 个数任意分成两组，每个数都可以选择分到第一组或分到第二组，有两种选择，

所以共有 2^7 = 128 种不同的分法。

    由于每组至少有一个数，所以还要将 {1,2,3,4,5,6,7}∪{ }  ，{ }∪{1,2,3,4,5,6,7}

这两种情形除去，因此，总的分法数应该是 128 - 2 = 126 。

    要使得分成两组数字之和相等，经过搜索，发现有且仅有下列 8 种情形：

        {1,2,4,7}∪{3,5,6} ，{3,5,6}∪{1,2,4,7} ，

        {1,2,5,6}∪{3,4,7} ，{3,4,7}∪{1,2,5,6} ，

        {1,3,4,6}∪{2,5,7} ，{2,5,7}∪{1,3,4,6} ，

        {2,3,4,5}∪{1,6,7} ，{1,6,7}∪{2,3,4,5} 。

    所以，两组数字之和相等的概率为 8/126 = 4/63 。

图老师2

7
21
35
63
[(1,), (2,), (3,), (4,), (5,), (6,), (7,)]
[(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (3, 4), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7), (5, 6), (5, 7), (6, 7)]
[(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 2, 7), (1, 3, 4), (1, 3, 5), (1, 3, 6), (1, 3, 7), (1, 4, 5), (1, 4, 6), (1, 4, 7), (1, 5, 6), (1, 5, 7), (1, 6, 7), (2, 3, 4), (2, 3, 5), (2, 3, 6), (2, 3, 7), (2, 4, 5), (2, 4, 6), (2, 4, 7), (2, 5, 6), (2, 5, 7), (2, 6, 7), (3, 4, 5), (3, 4, 6), (3, 4, 7), (3, 5, 6), (3, 5, 7), (3, 6, 7), (4, 5, 6), (4, 5, 7), (4, 6, 7), (5, 6, 7)]
(1, 6, 7)
(2, 5, 7)
(3, 4, 7)
(3, 5, 6)
----------------------------------------------------------------------------------------------------
0 0 4
两组数字之和相等的概率=4/63.

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python3.6程序运行 0.17190027236938477 秒。

图老师2

或：
7
21
35
35
21
7
126
[(1,), (2,), (3,), (4,), (5,), (6,), (7,)]
[(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (3, 4), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7), (5, 6), (5, 7), (6, 7)]
[(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 2, 7), (1, 3, 4), (1, 3, 5), (1, 3, 6), (1, 3, 7), (1, 4, 5), (1, 4, 6), (1, 4, 7), (1, 5, 6), (1, 5, 7), (1, 6, 7), (2, 3, 4), (2, 3, 5), (2, 3, 6), (2, 3, 7), (2, 4, 5), (2, 4, 6), (2, 4, 7), (2, 5, 6), (2, 5, 7), (2, 6, 7), (3, 4, 5), (3, 4, 6), (3, 4, 7), (3, 5, 6), (3, 5, 7), (3, 6, 7), (4, 5, 6), (4, 5, 7), (4, 6, 7), (5, 6, 7)]
[(1, 2, 3, 4), (1, 2, 3, 5), (1, 2, 3, 6), (1, 2, 3, 7), (1, 2, 4, 5), (1, 2, 4, 6), (1, 2, 4, 7), (1, 2, 5, 6), (1, 2, 5, 7), (1, 2, 6, 7), (1, 3, 4, 5), (1, 3, 4, 6), (1, 3, 4, 7), (1, 3, 5, 6), (1, 3, 5, 7), (1, 3, 6, 7), (1, 4, 5, 6), (1, 4, 5, 7), (1, 4, 6, 7), (1, 5, 6, 7), (2, 3, 4, 5), (2, 3, 4, 6), (2, 3, 4, 7), (2, 3, 5, 6), (2, 3, 5, 7), (2, 3, 6, 7), (2, 4, 5, 6), (2, 4, 5, 7), (2, 4, 6, 7), (2, 5, 6, 7), (3, 4, 5, 6), (3, 4, 5, 7), (3, 4, 6, 7), (3, 5, 6, 7), (4, 5, 6, 7)]
[(1, 2, 3, 4, 5), (1, 2, 3, 4, 6), (1, 2, 3, 4, 7), (1, 2, 3, 5, 6), (1, 2, 3, 5, 7), (1, 2, 3, 6, 7), (1, 2, 4, 5, 6), (1, 2, 4, 5, 7), (1, 2, 4, 6, 7), (1, 2, 5, 6, 7), (1, 3, 4, 5, 6), (1, 3, 4, 5, 7), (1, 3, 4, 6, 7), (1, 3, 5, 6, 7), (1, 4, 5, 6, 7), (2, 3, 4, 5, 6), (2, 3, 4, 5, 7), (2, 3, 4, 6, 7), (2, 3, 5, 6, 7), (2, 4, 5, 6, 7), (3, 4, 5, 6, 7)]
[(1, 2, 3, 4, 5, 6), (1, 2, 3, 4, 5, 7), (1, 2, 3, 4, 6, 7), (1, 2, 3, 5, 6, 7), (1, 2, 4, 5, 6, 7), (1, 3, 4, 5, 6, 7), (2, 3, 4, 5, 6, 7)]
(1, 6, 7)
(2, 5, 7)
(3, 4, 7)
(3, 5, 6)
(1, 2, 4, 7)
(1, 2, 5, 6)
(1, 3, 4, 6)
(2, 3, 4, 5)
----------------------------------------------------------------------------------------------------
0 0 4 4 0 0
两组数字之和相等的概率=8/126.

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python3.6程序运行 2.284693479537964 秒。

王守恩

将 1,2,3,4,5,6,7 任意分成两组，每组至少有一个数，求两组数字之和相等(即和都是14)的概率
简单一点：只要讨论每组有 1(对应6),2(对应5),3(对应4)个数就可以，
每组有 1 个数：可以有 7 种分法，其中和是14的有0种分法，
每组有 2 个数：可以有21种分法，其中和是14的有0种分法，
每组有 3 个数：可以有35种分法，其中和是14的有4种分法，
合计有7+21+35=63种分法，其中和是14的有0+0+4=4种分法。