Motivated by Chvátal's conjecture and Tomaszewaki's conjecture, we investigate the extreme value problem of two probability functions for the Gamma distribution. Let $α,β$ be arbitrary positive real numbers and $X_{α,β}$ be a Gamma random variable with shape parameter $α$ and scale parameter $β$. We study the extreme values of functions $P\{X_{α,β}\le E[X_{α,β}]\}$ and $P\{|X_{α,β}-E[X_{α,β}]|\le \sqrt{{\rm Var}(X_{α,β})}\}$. Among other things, we show that $ \inf_{α,β}P\{X_{α,β}\le E[X_{α,β}]\}=\frac{1}{2}$ and $\inf_{α,β}P\{|X_{α,β}-E[X_{α,β}]|\le \sqrt{{\rm Var}(X_{α,β})}\}=P\{|Z|\le 1\}\approx 0.6826$, where $Z$ is a standard normal random variable.