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古典概型是初等概率论中最基本的内容之一,解决古典概型题一般分两个步骤:第一步是选取适当的样本空间Ω,使它满足有限、等可能的要求,且把所研究事件 A 表示为样本空间Ω的某个子集;第二步则是计算样本点总数及基本事件数,在这一步中因为要计数,使得排列组合显得尤为重要.因此,大部分同学往往过于重视第二步而忽视了第一步,片面地认为计算概率必用排列组合知识.事实上,如果能合理地选取样本空间,对概率的性质予以充分的重视,复杂的排列组合的计算往往是可以简化,甚至能够避免的.本文将通过一些实例着重谈谈重视第一步即合理选择样本空间对解决古典概型问题的实际意义.
Classical probability model is one of the most basic contents of elementary probability theory. Solving the classical probability model problem is generally divided into two steps: The first step is to select an appropriate sample space Ω so that it satisfies the limited and equal possible requirements, and Research event A is expressed as a subset of the sample space Ω; the second step is to calculate the total number of sample points and the number of basic events. In this step, because of the count, the permutation and combination are particularly important. Therefore, most students often attach too much importance. The second step ignores the first step and unilaterally considers that the computational probability must be used to arrange combinatorial knowledge. In fact, if the sample space can be reasonably selected, sufficient attention is paid to the nature of the probability, and the calculation of complex permutations and combinations can be simplified. It can even be avoided. This article will focus on the actual significance of the first step, that is, the reasonable choice of sample space to solve the classical model problem through some examples.