"two N draw i") serves as a way to enumerate the different ways (or permutations) of obtaining i As in a sample of 2N.

Applying the binomial to the micro-centrifuge tube sampling results from the last section will illustrate how the binomial provides the probability for a specific sampling outcome. In the beaker, the blue and clear tubes were both at a frequency of p = q = 1/2. In the draws of N = 4, a result of two blue and two clear tubes occurred in four out of 10 draws or 40% of the time (Table 3.1). When drawing samples of tubes there are 2N possible combinations. So, for samples of N = 4 there are 24 = 16 combinations (like the number of genotypes in a Punnett square for four alleles at a locus). Of these 16 combinations, there are exactly six (bbcc, bcbc, bccb, cbcb, cbbc, ccbb) which yield two blue and two clear tubes. This same result can be obtained by using

to enumerate the number of possible permutations of outcomes of one type in a sample of 2N objects.

The ! notation stands for factorial, and n! equals 1 x 2 x 3 x ... x n - 1 x n and 0! = 1. The other part of the binomial formula calculates the probability of obtaining a sample of i blue tubes and 2N - i clear tubes. The blue tubes are at a frequency of p in the population, so the probability of sampling i of them is pi since each is an independent event. The same logic applies to the clear tubes, whose frequency in the population is q and the number sampled is the remaining sample size not made up of blue tubes or 2N - i, to give a probability of q2N-'. Bringing both of these components of the binomial formula together,

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