Protein Engineering Protocols - Mycobacteriology research center

142 Denault and Pelletierprobability that at least one theoretical variant has not been sampled is approx1–e –0.00206 =0.00206 or 0.2%. In this case, there is a greater than 99% probability(specifically, 100% – 0.206% = 99.794%) that all of the theoretical variantsn have been sampled (the fact that λ and the probability 1 – e –λ are veryclose numbers is not surprising, given that 1 + x e x for small x.).For our main example (n = 1 × 10 6 ; m = 2 × 10 7 ), the probability that fewer than50 from among the n theoretical variants have not been sampled in the m selecteditems is computed as the summation of the probabilities that each of 0, 1, 2, … upto 49 theoretical variants have not been sampled, yielding approx 78%. Thus:Similarly, the probability that fewer than 60 have not been sampled is approx98%. For ease of computation, use the Excel computation sheets (see Heading 2.and Fig. 1).3.1.1.3. PROBLEM C49−0. 00206ke 0.00206∑ = 078 .k!k=0How many times can we expect a variant i to appear in the sample? What isthe probability that a variant i appears 10 times? Three times? What about theprobability that the variant i appears 10 times or less?The probability that a variant appears a certain number of times is approximatedthrough a Poisson distribution (see Note 5) with parameter:λ= m n(3)so that the number of times we expect a certain variant i to appear is:Expected number of times the variant i occurs = λ= m nand the probability is:Probability (variant i occurs r times in thesampleofsize m)=er−λ λr!(4)(see Notes 2, 3, and 6). Be aware that the parameter of the Poisson distributionfor this problem, given in Eq. 3, is different from that used in Problems A andB, Subheadings 3.1.1.1. and 3.1.1.2., respectively, given in Eq. 1.The probability that a variant appears a certain number of times or less canbe computed by using Eq. 4 repetitively.For example, we take again n = 1 × 10 6 and m = 1 × 10 7 . We expect to find7variant i1×10= 10 times in the sample. The probability that i appears 10 times is:61×10