The probability that the number of successful of Bernoulli experiments are always close to their mean Given an infinite series of Bernoulli experiments, with probability p for the success of each, I wish to show that for every $p'<p$, the probability that at any point in the sequence more than $p'$ of the tests were successful is positive.
I am not entirely certain this theorem holds, but it feels intuitively true to me, yet I haven't found a way to prove it.
 A: At each time $n$, add $1-p'$ if the $n$th test is succesful, and $-p'$ otherwise. Call $X_n$ the position after $n$ steps. Since each test is succesful with probability $p$, the process $(X_n)$ performs a random walk with drift (the mean displacement) $(1-p')p+(-p')(1-p)=p-p'$. This drift is positive hence the probability that $X_n\gt0$ for every $n\geqslant1$ is positive.

Example: Assume that $p'=\frac13\lt p$, then $Y_n=3X_n$ is a $+2/-1$ random walk. Let $t_i$ denote the probability that $(Y_n)$ hits $0$ starting from $i$, then the probability that $X_n\gt0$ for every $n\geqslant1$ is $z=p\cdot(1-t_2)$. 
Furthermore, the Markov property after one step of the random walk shows that $(t_i)$ solves the system $t_0=1$, $t_i=pt_{i+2}+qt_{i-1}$ for every $i\geqslant1$, where $q=1-p$. 
Thus $t_i=ar^i+bs^i+ct^i$ for some $(a,b,c)$, where $(r,s,t)$ solve the characteristic equation $x=px^3+q$. Thus, $t=1$ and $(r,s)$ solve $px^2+px=q$, that is, $s=-\frac12-\frac12\sqrt{1+4q/p}$ and $r=-\frac12+\frac12\sqrt{1+4q/p}$, hence $s\lt-1$ and $0\lt r\lt1$. 
One knows that $t_i\to0$ when $i\to\infty$ hence $b=c=0$ and $t_i=r^i$ for every $i\geqslant0$. In particular, $z=p(1-r^2)=\frac32p-1+\frac12p\sqrt{1+4q/p}$.
Sanity checks: $z\to1$ when $p\to1$ (why?), $z\to0$ when $p\to\frac13$ (why?), and $z$ is a nondecreasing function of $p$ (why?).
