Some distributions and expectations - simplified

The purpose of the present paper is, first, to show how the binomial and the hypergeometric distributions could be lent a comparable form. The suggested presentation exhibits the similarity in the structure of the two distributions in accordance with the similarity in the verbal definitions of the random variables. Likewise, the minor dissimilarity in the two formulas reflects the difference in the respective verbal definitions. Next, the law of addition of expectations will be applied in order to compute the expectations of the above two distributions. The two expectations will be shown to be equal and the computation rendered simple. Finally, the power of the addition technique will be illustrated by computing the expectation for the number of runs in a binary random sequence. In an extended application, the expectation of the number of alternations in a random binary two-dimensional table will be computed, bypassing the complex problem of the distribution of that random variable.