  In this paper, we explore two conjectures about Rademacher sequences. Let
$(\epsilon_i)$ be a Rademacher sequence, i.e., a sequence of independent
$\{-1,1\}$-valued symmetric random variables. Set
$S_n=a_1\epsilon_1+\cdots+a_n\epsilon_n$ for $a=(a_1,\dots,a_n)\in
\mathbb{R}^n$. The first conjecture says that $P\ (\ |S_n\ |\leq \|a\|\
)\geq\frac{1}{2}$ for all $a\in \mathbb{R}^n$ and $n\in \mathbb{N}$. The second
conjecture says that $P\ (\ |S_n\ |\geq\|a\|\ )\geq \frac{7}{32}$ for all $a\in
\mathbb{R}^n$ and $n\in \mathbb{N}$. Regarding the first conjecture, we present
several new equivalent formulations. These include a topological view, a
combinatorial version and a strengthened version of the conjecture. Regarding
the second conjecture, we prove that it holds true when $n\leq 7$.

Probability;Combinatorics