Читать книгу Probability and Statistical Inference - Robert Bartoszynski - Страница 69

Consider two operations of some sort, which can be performed one after another. Leaving the notion of “operation” vague at the moment, we can make two assumptions:

1 The first operation can be performed in different ways.

2 For each of the ways of performing the first operation, the second operation can be performed in ways.

We have the following theorem:

Theorem 3.2.1 Under assumptions 1 and 2, a two‐step procedure consisting of a first operation followed by the second operation can be performed in distinct ways.

Proof: Observe that each way of performing the two operations can be represented as a pair with and , where is the th way of performing the first operation and is the th way of performing the second operation if the first operation was performed in th way. All such pairs can be arranged in a rectangular array with rows and columns.

We will now show some applications of Theorem 3.2.1.