本件は、samples=<十分な2の冪> にする手もあると思います。
%#!lualatex
\documentclass[a5paper,landscape]{article}\pagestyle{empty}
\usepackage{tikz}
\begin{document}
%% 第一事象と第二事象をあわせる
\begin{tikzpicture}[scale=1]
\draw[->,>=stealth] (-1.5,0)--(1.5,0) node[right]{$x$};
\draw[->,>=stealth] (0,-0.5)--(0,1.5) node[left]{$y$};
% % \draw[domain=-1:1, samples=300] plot(\x, {sqrt(1-(\x)^2)});
\draw[domain=-1:0.01, samples=300] plot(\x, {sqrt(1-(\x)^2)});
% % \draw[domain=1:-1, samples=300] plot(\x, {sqrt(1-(\x)^2)});
\draw[domain=1:-0.01, samples=300] plot(\x, {sqrt(1-(\x)^2)});
\end{tikzpicture}
%% samples=<2の冪>
\begin{tikzpicture}[scale=1]
\draw[->,>=stealth] (-1.5,0)--(1.5,0) node[right]{$x$};
\draw[->,>=stealth] (0,-0.5)--(0,1.5) node[left]{$y$};
\draw[domain=-1:1, samples=512] plot(\x, {sqrt(1-(\x)^2)});
\end{tikzpicture}
\end{document}