挺哥布置的作业：

BreakDS posted @ Sun, 24 Aug 2008 18:08:31 +0800 in Uncategorized , 1286 readers

我只拣软柿子捏....

第七集:仙剑4的变态攻击

韩凌纱拿的武器一般是二次攻击的,如果给她加上"武器复魔连击"的魔法效果后,她的攻击次数有时会很变态.因为这个魔法效果让她的每次攻击都有20%的概率多攻击4次,这多攻击的4次中也都有20%概率多攻击4次,如此下去.

 

试问:

1.该美女攻击次数的数学期望是多少?

2.该美女攻击次数的分布函数是什么?

3.如果20%这个参数和4这个参数有所改变,结论又如何?


答：先不考虑开始是二次攻击，只考虑为一次攻击，则整个攻击可以看成一棵特殊的四叉树。 称为“美女树”吧。 美女树的每个结点有4个孩子或者没有(叶子结点)。

        美女树的每个结点代表一次攻击。

        美女树的每个结点有4个孩子的概率是0.2，是叶子结点的概率为(1-0.2=0.8)

        于是：

1. 对于美女的第一次攻击为根结点的美女树，设总结点数期望为E。 又因为美女树的每个结点以下又是一棵美女树，于是有：

E = 0.8 \times 1 + 0.2 \times (E \times 4)

解得 E = 4

因为开始有两次攻击，两个总根结点，故为2E = 8。 我没有玩过仙剑，但是若是以3RD里我的理查德来看，他一次能2w，8次就是16w，期望16w的攻击任哪个最终boss都死了。。。

2. 注意每个结点的扩展概率为0.2。

对于一棵任意形状的树总有结点可以扩展(最郁闷也有一个总根结点)

一棵既定的树的任意两个叶子结点是否可以扩展是无关的(independent吧，防岐义)。

一个结点若扩展出4个孩子都没有继续扩展(4个公公^_^)的概率，就是4个不扩展的乘积，为 r = 0.8^4

称为断子绝孙概率。

还是先不考虑开始是2次攻击，于是：

P{ A = 1 } = 0.8

P{ A = 5 } = 0.2 * r

P{ A = t } = 叶子结点数(t-4) * P{ A = t-4 } /0.8 * 0.2 * r

叶子结点数很好算，每次扩展+3

然后就是傻算了。

因为是两次攻击，两个这样的分布加一下，因为是傻算。。 数学系没培养我们这个技能。。 这道题感觉是比较正的答案。

3. 代回上面两问吧。答案不一样，显然0.2越大越猛。

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