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In 1883, German Mathematician George Cantor introduced Cantor ternary set which is a self-similar fractal. K. J. Falconer (1990) defined random Cantor set with statistical self-similarity. The purpose of this paper is to introduce generalized random Cantor sets (one 5^<em>th</em>, one 7^<em>th</em> and in general one <em>p</em>^<em>th</em>). Some properties and results of random Cantor one <em>p</em>^<em>th</em> set have also been obtained. We compute Hausdorff dimension of random Cantor one <em>p</em>^<em>th</em> sets and show that Hausdorff dimension of these random Cantor sets is less than that of Hausdorff dimension of Cantor one <em>p</em>^<em>th</em> sets, calculated by Ashish et al. (2013).