扔硬币的问题（正正反和正反反）

终结了：
The problem is same as asking for the probability of seeing HHT before HTT when the current state is H because you need an H to jumpstart both sequences. Let such probability be P。
Current State: H
There are 2 cases:
1. next flip=H (with 1/2 probability). We have HH. No matter what the next flips are, we will always have a string of ,,,HHHHH(HH). We will always see HHT before HTT as the first T will complete the HHT sequence.
2. next flip=T (with 1/2 probability). We have HT.
If the next flip is T (total probability 1/2*1/2=1/4) then we see HTT before HHT.
If the next flip is H (total probability 1/4) then we get back to the original state H.
Therefore, P=1/2+1/4*0+1/4*P , or P=2/3

果然这是一个有名的问题（Penney's game）
http://en.wikipedia.org/wiki/Penney%27s_game

有一个圆桌。让你随便给它安装3条一样的桌腿，求这个圆桌能站立的概率是多少。
我觉得站立的条件是“圆桌的重心落在以3条腿为顶点的三角形内部”。

不知道我的解法对不对。欢迎拍砖。
1/2
设圆桌周长的1/2为L,那么第二条腿落的位置是距离第一条腿的L距离内。可在第一条腿的左边或右边，所以是对称的。设距离为X，那么第三条必然在“前两个点过圆心之后，与圆周围成的范围内”的时候，桌子才能站立。所以当第二条腿与第一条腿距离为X的时候，桌子能站立的概率是X/(2L).
所以我们得到积分：
积分结果是1/2

第三题：
http://math.stackexchange.com/questions/27524/fair-value-of-a-hat-drawing-game

第四题
有一个桶，里面有等量的红球和蓝球。取出3个，求红球个数是奇数的概率。取出10个，求红球个数是奇数的概率。
这个没想通怎么做。

第五题
两个普通的骰子一起扔，可得的点数之和有固定的概率分布。现在把每个骰子上的点数改成其他的正整数（两个骰子上的点可以不同），再扔，能否使点数和的概率分布不变？

@8826055 大神，你人呢？

The experiment is to toss a coin until we can see a pattern of HTH, for example HTTTHTTHTH or the pattern HTT such as THTHTHTHTT. We do this 100 times for HTH and another 100 times for HTT writing down the results each time so you can create the mean (average) number of toss before each pattern occurs. If you understand the process then pick a following statement based on what you think is true.
A. It will take longer (more tosses) for HTH to occur in random coin tosses than for HTT to occur.
B. They will take the same number of tosses for either HTH or HTT to occur.
C. It will be sooner (less tosses) for HTH to occur in a random coin tosses than for HTT to occur.
We assume that because the coin toss is random that everything that comes from it is also random. This ignores the fact that both HTT and HTH are not random; they are non-random patterns that have been created. Most people think that B is the correct answer, although it is actually A which is correct. The average number of tosses for HTH is 10 before the pattern occurs and 8 tosses before the pattern HTT occurs. To explain why A is correct; it has to do with the third toss in the patterns. When we are looking for the pattern HTH, and we have the HT the next toss could result in H in which case we complete the pattern or T where we have to start over again. When we are looking for the pattern HTT, and we have HT the next toss could result in a T in which case we complete the pattern or a H which would not complete the pattern but it is then possible to use that H in the start of the next HTT. It’s weighted for the benefit of HTT and that is why we need fewer coin tosses to get the pattern HTT.

第六题
We roll a 3 sided dice 100 times. The faces have values 1, 2 and 3. What is
the probability that no 3 consecutive rolls add up to 4?
我本来想解出递推公式的。比如： f(n)=a*f(n-1)+b*f(n-2)+...... （f(n)代表总共扔n次，存在连续扔3次的和是4的概率）。结果我失败了。。。

第八题：
You have a deck of cards, 26 red, 26 black. These are turned over, and at any point you may stop and exclaim "The next card is red.". If the next card is red you win £10. What's the optimal strategy? Prove this is the optimal strategy.
一副扑克，26红，26黑。混合后背面向上，变成一摞。每轮翻开最上面一张。在任意一轮开始之前，你都可以叫停，声明“这轮翻开的是红牌”。如果这轮翻开的真是红牌，你赢钱。 你的最优策略是什么？怎么证明？

第九题：
扔一个普通的骰子,每一次都得到一个点数（1-6）。总共扔K次。假如这K次的点数乘积大于100，你赢。否则我赢。那么K应该等于几，这个游戏才能使我们的胜率大致相等（fair game）？

第十一题：
已知:有若干红球黑球在袋中，每次取一个球并放回，前１０次都是黑球。
问第１１次是黑球的概率。
我看到比较有用的讨论是：
A.11/12
B.可以解释下吗？为什么和袋子里总共的球数目没关系？
如果数目很大，用积分近似算出来是11/12，但是如果很少，比如只有2个，好像应该更
接近1啊
C.If I am a bayesian, the prob is 1 even only after the first black ball is
drawn. You can see it by maximum likelihood.The prob stays 1 until some
other events coming up to modify it.