In many circumstances, given an ordered sequence of one or more types of elements or symbols, the objective is to determine the existence of any randomness in the occurrence of one specific element, say type 1. This method can help detect non-random patterns, such as wins or losses in a series of games. Existing methods of tests based on total number of runs or tests based on length of longest run (Mosteller (1941)) can be used for testing the null hypothesis of randomness in the entire sequence, and not a specific type of element. Moreover, the Runs Test often yields results that contradict the patterns visualized in graphs showing, for instance, win proportions over time. This paper develops a test approach to address this problem by computing the gaps between two consecutive type 1 elements, by identifying patterns in occurrence and directional trends (increasing, decreasing, or constant), applies the exact Binomial test, Kendall's Tau, and the Siegel-Tukey test for scale problems. Further modifications suggested by Jan Vegelius(1982) have been applied in the Siegel Tukey test to adjust for tied ranks and achieve more accurate results. This approach is distribution-free and suitable for small sample sizes. Also comparisons with the conventional runs test demonstrates the superiority of the proposed method under the null hypothesis of randomness in the occurrence of type 1 elements.

翻译：在许多情况下，给定一个由一种或多种元素或符号构成的有序序列，目标在于判断其中某一特定元素（例如类型1）的出现是否存在随机性。该方法可用于检测非随机模式，例如一系列游戏中的胜负记录。现有检验方法（如基于游程总数的检验或基于最长游程长度的检验（Mosteller (1941)）可用于检验整个序列的随机性原假设，但无法针对特定类型元素进行检验。此外，游程检验的结果常与可视化图表（例如随时间变化的胜率图）所显示的模式相矛盾。本文通过计算两个连续类型1元素之间的间隔，识别其出现模式与方向性趋势（递增、递减或恒定），并应用精确二项式检验、Kendall's Tau检验以及针对尺度问题的Siegel-Tukey检验，构建了一种解决该问题的检验方法。在Siegel-Tukey检验中采用了Jan Vegelius（1982）提出的修正方案，以处理结秩问题并获得更准确的结果。该方法属于无分布检验，适用于小样本场景。与常规游程检验的比较表明，在类型1元素出现具有随机性的原假设下，本方法具有优越性。