The probability of an event is in the range of [0, 1]. In a sample space S, the value of probability determines whether an outcome is true or false. The probability of an event Pr(A) that will never occur = 0. The probability of the event Pr(B) that will certainly occur = 1. This makes both events A and B thus a certainty. Furthermore, the sum of probabilities Pr(E1) + Pr(E2) + ... + Pr(En) of a finite set of events in a given sample space S = 1. Conversely, the difference of the sum of two probabilities that will certainly occur is 0. Firstly, this paper discusses Bayes' theorem, then complement of probability and the difference of probability for occurrences of learning-events, before applying these in the prediction of learning objects in student learning. Given the sum total of 1; to make recommendation for student learning, this paper submits that the difference of argMaxPr(S) and probability of student-performance quantifies the weight of learning objects for students. Using a dataset of skill-set, the computational procedure demonstrates: i) the probability of skill-set events that has occurred that would lead to higher level learning; ii) the probability of the events that has not occurred that requires subject-matter relearning; iii) accuracy of decision tree in the prediction of student performance into class labels; and iv) information entropy about skill-set data and its implication on student cognitive performance and recommendation of learning [1].

翻译：事件的概率取值范围为[0,1]。在样本空间S中，概率值决定了结果的真假性。必然不发生的事件A的概率Pr(A)=0，必然发生的事件B的概率Pr(B)=1，这使得事件A和B均成为确定性事件。此外，给定样本空间S中有限事件集的概率之和Pr(E1)+Pr(E2)+...+Pr(En)=1。相应地，两个必然发生事件的概率之和的差值为0。本文首先讨论贝叶斯定理，随后探讨学习事件发生概率的互补性与差异性，并将其应用于学生学习对象的预测。基于总概率为1的前提，本文提出：argMaxPr(S)与学生表现概率之差可量化学习对象对学生的权重。通过技能集数据集的计算机实现流程，本文展示了：i) 已发生的高阶学习技能集事件概率；ii) 未发生且需重新学习学科内容的技能事件概率；iii) 决策树对学生成绩进行类别标签预测的准确性；iv) 技能集数据的信息熵及其对学生认知表现与学习推荐的启示[1]。