This paper will establish Chebyshev-Jensen-type inequalities which involving the Chebyshev products as follows:\[\langle X^{1}, X^{2}, \ldots, X^{m}\rangle_{\chi}\geq \left[ X^{1}, X^{2}, \ldots, X^{m}\right]_{\chi}\geq \chi\left( \overline{X^{1}}, \overline{X^{2}}, \ldots, \overline{X^{m}}\right)\]and\[\langle f_{1}, f_{2}, \ldots, f_{m}\rangle_{\chi}\geq \left[f_{1}, f_{2}, \ldots,f_{m}\right]_{\chi}\geq \chi\left( \overline{f_{1}}, \overline{f_{2}}, \ldots, \overline{f_{m}}\right),\]as well as display their applications in probability theory and surround system, especially, the observation angles inequalities are obtained as follows:\[1\geq \frac{1}{n}\sum_{i=1}^{n}P\left(R_{i}\right)\geq \frac{1}{n^{m}}\sum_{1\leq i_{1},\ldots,i_{m}\leq n}P(R_{i_{1},\ldots,i_{m}})\geq\left(\frac{2}{n}\right)^{m+|\alpha|}.\]The research methods of the paper are based on the mathematical induction, reorder method and the dimension reduction method.

2010 MSC: 26D15, 26E60, 51K05, 52A40