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Problems on the uniform distribution

(1) suppose U is a continuous uniform [0,1] random variable. what is the
probability that the decimal expansion of U contains no fives?

(2) What is the ...


Find the missing numbers

Given an array of size n. It contains numbers in the range 1 to n. Each
number is present at least once except for 2 numbers. Find the missing
numbers.


Neighbourly Polyhedra

Can you construct 1000 internally disjoint convex polyhedra in R^3 any two of which share a face?

Tags: math geometry

擂台赛的策略

假设两队人马进行擂台比赛。开始两队各派一人比赛,获胜者留在台上,失败者所在一方再出一人于台上的人比赛,而且失败者以后不能再上台。如此下去,直到一方派不出人比赛时该方失败。

假设每个人的能力用一个正数表示。每场比赛的结果是互相独立的随机事件,能力a的人战胜能力b的人的概率永远是a/(a+b)。

现在一队永远派最强的人上场,这是最佳策略吗?如果你是二队的教练,你会采取什么策略?


Loaded Dice to Make Equally Likely Sums

We have two loaded dice, with 1,2,...,6 coming up with various (possibly different) probabilities on each. Is it possible that when we roll them both, each of the sums 2, ...