Probability and Statistical Inference. Robert Bartoszynski

Probability and Statistical Inference

Probability and Statistical Inference - Robert Bartoszynski

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alt="images"/>. Then

equation

      Separating the term for images in the first sum, and the term for images in the last sum, we may write

equation

      Theorem 3.3.4 The binomial coefficients satisfy the identities

       and

      (3.18)equation

      We also have the following theorem:

      Theorem 3.3.5 For every images and every images

      while the left‐hand side equals

      As a consequence of (3.19), we obtain a corollary:

      Corollary 3.3.6

equation

      Proof: Take images in (3.19) and use the fact that

equation

      Below we present some examples of the use of binomial coefficient in solving various probability problems, some with a long history.

      

      Example 3.7

      Let us consider a selection without replacement from a finite set containing two categories of objects. If images balls are to be selected from an urn containing images red and images blue balls, one might want to know the probability that there will be exactly images red balls chosen.

      Solution

      We apply here the “classical” definition of probability. The choice of images objects without replacement is the same as choosing a subset of images objects from the set of total of images objects. This can be done in images different ways. Since we must have images red balls, this choice can be made in images ways. Similarly, images blue balls can be selected in images ways. As each choice of images red balls can be combined with each of the images choices of blue balls then, by Theorem 3.2.2, the total number of choices is the product images and

      

      Example 3.8