Allele Coverage - missing in all sites we have L N  ...

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This supplementary note is to further explain how to get the minimum population size to achieve certain level of allele coverage in more details (page 26-27 of Part 4). Given a population of chromosomes (say N chromosomes), and the length of chromosome is L , For a binary chromosome, Chromosome 1 Chromosome 2 : : : : Chromosome N The probability of having no ‘1’ in the first MSB in chromosome 1 is ½. Similarly, the probability of having no ‘1’ in the first MSB in all N chromosomes are N 2 1 . Similarly, to have no ‘0’ is also N 2 1 . Therefore, to have no gene missing is 1 2 1 1 2 1 2 1 1 - - = - - N N N in a particular site (bit position in a chromosome). Since there are L sites (length of the chromosome), to have no gene
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Unformatted text preview: missing in all sites, we have L N              --1 2 1 1 . For high cardinality alphabet (eg. for hexadecimal, K=16; for octal, K=8) the probability to have no missing gene in a site (alphabet position) is assignment possible All positions N to symbols K of assignment possible where there are K!S(N,K) possible assignments of K symbols to N positions and the number of all possible assignments is K N (as you can place any one symbol in N positions). To have no gene missing in all L sites (length of the chromosome), we have ( ) L N K K N S K       , ! ....
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