The theoretical investigation of Evolutionary Algorithms (EAs) has increased our understanding of the computational mechanism of algorithms. OneMax is a test function most frequently and deeply studied in the field of EAs. In this work, a method is presented for describing the runtime properties of $(1+1)$ EA on OneMax. This method is motivated by the work of Erdös and Rényi treating the coupon collector's problem. They showed that the success probability of the coupon collector's problem is given by a function of double exponential form, and that the number of uncollected coupons follows the Poisson distribution. Today, the double exponential function is called Gumbel function, which is one of three fundamental functions in extreme value statistics. We introduce an algorithm that is a variant of the $(1+1)$ EA, First Order Evolutionary Algorithm (FO-EA). FO-EA takes into account only the effect of single-bit mutations in the $(1+1)$ EA, which in general includes multiple-bit mutations. We modified the method of Erdös and Rényi to apply FO-EA. We apply the Gumbel distribution for calculating the success probability of the $(1+1)$ EA on OneMax. This method turns out to give a sufficiently reliable estimation for success probabilities, even in the tail region.

进化算法（EAs）的理论研究增加了我们对算法计算机制的理解。OneMax是EA领域中最频繁且深入研究的测试功能。在这项工作中，提出了一种用于描述运行时属性的方法。$（\mathrm{1个}+\mathrm{1个}）$EA在OneMax上。这种方法是由Erdös和Rényi处理优惠券收集者的问题推动的。他们表明，优惠券收集者问题的成功概率是由双指数形式的函数给出的，未领取的优惠券的数量遵循泊松分布。今天，双指数函数称为Gumbel函数，它是极值统计中的三个基本函数之一。我们介绍了一种算法，该算法是$（\mathrm{1个}+\mathrm{1个}）$EA，一阶进化算法（FO-EA）。FO-EA仅考虑了单位突变在$（\mathrm{1个}+\mathrm{1个}）$EA，通常包括多位突变。我们修改了Erdös和Rényi的方法以应用FO-EA。我们应用Gumbel分布来计算成功的概率$（\mathrm{1个}+\mathrm{1个}）$EA在OneMax上。结果证明，即使在尾部区域，该方法也可以对成功概率给出足够可靠的估计。