1 Business School, Chengdu University, Chengdu, 610106 China.
2 College of Mathematics, Sichuan University, Chengdu, 610065 China.
3 Department of Mathematics and Statistics, Concordia University, Montreal, H3G 1M8 Canada.

Let a, ß be positive real numbers and let $X_{\alpha ,\beta }$ be a Gamma random variable with shape parameter a and scale parameter ß. We study infimum values of the function $(\alpha ,\beta )\mapsto P\{X_{\alpha ,\beta }\le \kappa E[X_{\alpha ,\beta }\left]\right\}$ for any fixed $\kappa >0$ and the function $(\alpha ,\beta )\mapsto P\left\{\right|X_{\alpha ,\beta }-E\left[X_{\alpha ,\beta }\right]|\le \sqrt{Var\left(X_{\alpha ,\beta }\right)}\}$. We show that ${inf}_{\alpha ,\beta }P\{X_{\alpha ,\beta }\le E[X_{\alpha ,\beta }\left]\right\}=\frac{1}{2}$ and ${inf}_{\alpha ,\beta }P\left\{\right|X_{\alpha ,\beta }-E\left[X_{\alpha ,\beta }\right]|\le \sqrt{Var\left(X_{\alpha ,\beta }\right)}\}=P\left\{\right|Z|\le 1\}\approx 0.6827$, where Z is a standard normal random variable.