h*****a
发帖数: 52 | 1 1. There are some unbiased coins (equal probability of having head and tail
when tossed). Initially, all the coins are tossed. The following steps are
followed after that:
- turn all the Head up coins to Tail.
- toss all the Tail up coins.
Find the probability of finding the a coin Head up after 'n' pairs of steps,
( or to frame it differently, find the number of coins with Head up) (Hint:
think in terms of ratio)
Ideas? | s*******s
发帖数: 1568 | 2 没懂，
每次都全部重投？
tail
steps,
Hint: | n****e
发帖数: 629 | 3 第二步是不包括第一步turn的那些?不然trivial啊……
tail
steps,
Hint:
【在 h*****a 的大作中提到】
: 1. There are some unbiased coins (equal probability of having head and tail
: when tossed). Initially, all the coins are tossed. The following steps are
: followed after that:
: - turn all the Head up coins to Tail.
: - toss all the Tail up coins.
: Find the probability of finding the a coin Head up after 'n' pairs of steps,
: ( or to frame it differently, find the number of coins with Head up) (Hint:
: think in terms of ratio)
: Ideas?
| n****e
发帖数: 629 | 4 如果这样也好算吧
写个转移矩阵就可以了吧
【在 n****e 的大作中提到】
: 第二步是不包括第一步turn的那些?不然trivial啊……
:
: tail
: steps,
: Hint:
| h*****a
发帖数: 52 | 5 对，当然不包括
【在 n****e 的大作中提到】
: 第二步是不包括第一步turn的那些?不然trivial啊……
:
: tail
: steps,
: Hint:
| s*******s
发帖数: 1568 | 6 Markov chain?
Let Sn to be state with n heads up and P is the transition prob,
the finial distribution is Sn*P^m
【在 h*****a 的大作中提到】
: 对，当然不包括
| s*******s
发帖数: 1568 | 7 写了个R程序，
n = 100
m = 5
p = 0.5
v1= 0:n
s0=dbinom(v1,n,p)
Pmatrix = matrix(0, nrow = n+1, ncol = n+1)
for (i in 0:n) {
v1 = 0:i
Pmatrix[i+1,1:(i+1)] = dbinom(v1,i,p)
}
for (i in 1:m) {
s0=s0%*%Pmatrix
}
E = 0;
for (i in 0:n) {
E = E + s0[i+1]*i
}
print(E)
【在 s*******s 的大作中提到】
: Markov chain?
: Let Sn to be state with n heads up and P is the transition prob,
: the finial distribution is Sn*P^m
| n****e
发帖数: 629 | 8 呃呃呃
不能diagonalize一下然后写通式么……
【在 s*******s 的大作中提到】
: 写了个R程序，
: n = 100
: m = 5
: p = 0.5
: v1= 0:n
: s0=dbinom(v1,n,p)
: Pmatrix = matrix(0, nrow = n+1, ncol = n+1)
: for (i in 0:n) {
: v1 = 0:i
: Pmatrix[i+1,1:(i+1)] = dbinom(v1,i,p)
| J****g
发帖数: 103 | 9 这个最后要找的不就是投n次， 次次为正面的概率吗？ 还是我理解错了？
【在 s*******s 的大作中提到】
: Markov chain?
: Let Sn to be state with n heads up and P is the transition prob,
: the finial distribution is Sn*P^m
| y**k
发帖数: 222 | | p*****k
发帖数: 318 | 11 just need to consider one coin, since each coin is independent.
one can get (the method is outlined above):
P(H)=1/3+1/6*(-1/2)^n, P(T)=1-P(H)
with the limiting distribution {1/3,2/3} when n is large. |
|
