b******y
发帖数: 126 | 1 A gambler bet on a fair game, every time he bets $1, he has $10 before the
game, if he loses all his money, the game ends,and he can only play 50 times
, what is the probability he will lose all his money. | s*******n
发帖数: 66 | 2 some rough idea
use random walk and stopping time
the approximate answer is 2*N(-sqrt(2))
hope big bull to verify or correct it
times
【在 b******y 的大作中提到】
: A gambler bet on a fair game, every time he bets $1, he has $10 before the
: game, if he loses all his money, the game ends,and he can only play 50 times
: , what is the probability he will lose all his money.
| a****h
发帖数: 126 | 3 (C(50,0)+C(50,1)+...+ C(50,20))*2/(2^50) ?
times
【在 b******y 的大作中提到】
: A gambler bet on a fair game, every time he bets $1, he has $10 before the
: game, if he loses all his money, the game ends,and he can only play 50 times
: , what is the probability he will lose all his money.
| A**u
发帖数: 2458 | 4 这玩意就是随机游走。
玩50次，就是50步， x1+x2+...+x50
初始是X0=10, 只要这个随机游走碰到 0 就结束
所以，等价于 X0 = 0, 50步的随机游走，碰到 -10的的情况
所有路径的总可能数目是 2^50
破产的路径数目 = 第50步以前破产的路径数目 + 第50步破产
用Reflection Principle
地50步以前破产的路径数目= 2 * (第50步到达-12的路径+第50步到达-14+...
+第50步到达-50的路径)
第50步到达-12的路径数目 = 31个T,19个H = C(50,31)
第50步到达-14的路径数目 = 32个T,18个H = C(50,32)
刚好第50步破产的路径数目=30个T, 20个H, 且始终没有碰到-10
这个就是1个Ballot问题， 相当于a=30,b=20,
路径数目=(a-b)/(a+b)*C(a+b,a)
然后就可以算概率了
times
【在 b******y 的大作中提到】
: A gambler bet on a fair game, every time he bets $1, he has $10 before the
: game, if he loses all his money, the game ends,and he can only play 50 times
: , what is the probability he will lose all his money.
| l*******1
发帖数: 113 | 5 monte carlo agrees with 0.1608 | A**u
发帖数: 2458 | 6 我上面算的， 应该是0.1273
【在 l*******1 的大作中提到】
: monte carlo agrees with 0.1608
| l*******1
发帖数: 113 | | f********g
发帖数: 22 | 8 How much does he get if he wins?
times
【在 b******y 的大作中提到】
: A gambler bet on a fair game, every time he bets $1, he has $10 before the
: game, if he loses all his money, the game ends,and he can only play 50 times
: , what is the probability he will lose all his money.
| A**u
发帖数: 2458 | 9 恩 上面计算算错了，应该是 0.1524
和你的值差不多
> c = choose(50,31)+choose(50,32)+choose(50,33)+choose(50,34)+choose(50,35)+
+ choose(50,36)+choose(50,37)+choose(50,38)+choose(50,39)+choose(50,40)+
+ choose(50,41)+choose(50,42)+choose(50,43)+choose(50,44)+choose(50,45)+
+ choose(50,46)+choose(50,47)+choose(50,48)+choose(50,49)+choose(50,50)
> x = 10/50*choose(50,20)
> (choose(50,20)-x)/2^50+2*c/n
[1] 0.1524078
【在 l*******1 的大作中提到】
: Try monte carlo :)
| n**l
发帖数: 17 | 10 30 down, 20 up的时候,程序算出来, hit -10的概率基本等于1
所以是,x=(2c+choose(50,20))/2^50
0.160779
)+
【在 A**u 的大作中提到】
: 恩 上面计算算错了，应该是 0.1524
: 和你的值差不多
: > c = choose(50,31)+choose(50,32)+choose(50,33)+choose(50,34)+choose(50,35)+
: + choose(50,36)+choose(50,37)+choose(50,38)+choose(50,39)+choose(50,40)+
: + choose(50,41)+choose(50,42)+choose(50,43)+choose(50,44)+choose(50,45)+
: + choose(50,46)+choose(50,47)+choose(50,48)+choose(50,49)+choose(50,50)
: > x = 10/50*choose(50,20)
: > (choose(50,20)-x)/2^50+2*c/n
: [1] 0.1524078
| | | A**u
发帖数: 2458 | 11 恩不错，我这里算 刚好 hit -10时候，算错了，
谢谢更正
【在 n**l 的大作中提到】
: 30 down, 20 up的时候,程序算出来, hit -10的概率基本等于1
: 所以是,x=(2c+choose(50,20))/2^50
: 0.160779
:
: )+
| r*******1
发帖数: 69 | 12 Algee. 在0处reflect, 刚才看错题了
【在 A**u 的大作中提到】
: 恩不错，我这里算 刚好 hit -10时候，算错了，
: 谢谢更正
| x******a
发帖数: 6336 | 13 Let T be the first time he loses all money, then
P(T<=50)=sum P(T<=50,S_50=n) , n is even, from -40 to 40
Next, if n<=0, P(T<=50, S_50=n)=P(S_50=n).
If n>0, P(T<=50,S_50=n)=P(S_50=-n).
times
【在 b******y 的大作中提到】
: A gambler bet on a fair game, every time he bets $1, he has $10 before the
: game, if he loses all his money, the game ends,and he can only play 50 times
: , what is the probability he will lose all his money.
| g******r
发帖数: 29 | 14 50 够大了 觉得可以用大数定理
估计是让你把其当作布朗运动 还有反射原理
times
【在 b******y 的大作中提到】
: A gambler bet on a fair game, every time he bets $1, he has $10 before the
: game, if he loses all his money, the game ends,and he can only play 50 times
: , what is the probability he will lose all his money.
| d*****o
发帖数: 34 | 15 Martingale
times
【在 b******y 的大作中提到】
: A gambler bet on a fair game, every time he bets $1, he has $10 before the
: game, if he loses all his money, the game ends,and he can only play 50 times
: , what is the probability he will lose all his money.
| I******4
发帖数: 5 | 16 we have the equivalent solution
【在 a****h 的大作中提到】
: (C(50,0)+C(50,1)+...+ C(50,20))*2/(2^50) ?
:
: times
| x********9
发帖数: 31 | 17 Reflection Principle.
Suppose the gambler can have negative wealth. After 50 rounds, the
distribution of his wealth is just a binomial distribution from -40 to 60.
That distribution can be approximated by a Gaussian with mean 10 and
variance 50. The probability of having a wealth lower than zero is
approximatively N(-10/sqrt(50)).
By reflection principle, the prhitting time is nothing else than 2*N(-10/
sqrt(50)). | f*******y
发帖数: 52 | 18 2*P(S_50<-10)+P(S_50=-10), S_n is the random work starting from 0. | r**a
发帖数: 536 | 19 你这个不对吧。reflection用在有drift的brownian motion上是要有变形的。不能直接
用。
.
【在 x********9 的大作中提到】
: Reflection Principle.
: Suppose the gambler can have negative wealth. After 50 rounds, the
: distribution of his wealth is just a binomial distribution from -40 to 60.
: That distribution can be approximated by a Gaussian with mean 10 and
: variance 50. The probability of having a wealth lower than zero is
: approximatively N(-10/sqrt(50)).
: By reflection principle, the prhitting time is nothing else than 2*N(-10/
: sqrt(50)).
| m***w
发帖数: 404 | 20 弱弱的问一下，为什么要乘2？reflection principles如何体现？
【在 A**u 的大作中提到】
: 恩不错，我这里算 刚好 hit -10时候，算错了，
: 谢谢更正
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