- #1
caffeinemachine
Gold Member
MHB
- 799
- 15
The following question came up when me and a friend of mine were discussing some basic things about probability:Let $p$ be a real number in $[0,1]$.
Does there exist a sequence $(x_1, x_2, x_3, \ldots)$ with each $x_i$ being either $0$ or $1$, such that
$$
\lim_{n\to \infty} \frac{f(n)}{n} =p
$$
where $f(n)= x_1+x_2+\cdots+x_n$, that is, $f(n)$ is the number of times $1$ has appeared in the first $n$ slots. Motivation: Consider a coin which may or may not be fair, and say the probability of "heads" showing up is $p$.
Suppose we want to have a machine which simulates this coin.
That is, we want to have a machine which shows either "H" or "T" every second ('second' here is a unit of time) on its screen.
If the machine properly simulates the coin, then we must have the average frequency of "H" occurring is $p$.
Does there exist a sequence $(x_1, x_2, x_3, \ldots)$ with each $x_i$ being either $0$ or $1$, such that
$$
\lim_{n\to \infty} \frac{f(n)}{n} =p
$$
where $f(n)= x_1+x_2+\cdots+x_n$, that is, $f(n)$ is the number of times $1$ has appeared in the first $n$ slots. Motivation: Consider a coin which may or may not be fair, and say the probability of "heads" showing up is $p$.
Suppose we want to have a machine which simulates this coin.
That is, we want to have a machine which shows either "H" or "T" every second ('second' here is a unit of time) on its screen.
If the machine properly simulates the coin, then we must have the average frequency of "H" occurring is $p$.