Statistical evaluation of the Luria - Delbruck thought experiment

  mean  = (x_i) / c
  variance ² =  [ - x_i]² / c
 
In 1943, variance is calculated by MOSSOM formula on a mechanical calculating machine

Both Thought Experiments involve 10 observed Ton^r cells our of 64 total cells

Suppose probability of induction (a)  = 10 random events / 64 cells = 15%
    observed = 3, 1, 5, & 1 = 10 Ton^r colonies      
       mean = (3 + 1 + 5 + 1) / 4 = 2.5 Ton^r  per culture
       variance = [(2.5 - 3)² + (2.5 - 1)² + (2.5 - 5)² + (2.5 - 1)²] / 4 = 2.75
                                                        or ( 3² + 1² + 5² + 1²) / 4  -  2.5²  = 2.75

        Induction Hypothesis predicts a Poisson Distribution for rare, random events: variance = mean


Suppose mutation rate (a) = 2 events / 60 cell divisions = 0.033 mutations / cell / generation
        observed = 2, 0, 8, & 0 = 10 Ton^r colonies
        mean  = (2 + 0 + 8 + 0) / 4 = 2.5 Ton^r as before
        earlier Ton^r mutations leave more offspring (as in Culture 3)
       variance =  [(2.5 - 2)² + (2.5 - 0)² + (2.5 - 8)² + (2.5 - 0)²] / 4 = 10.75
                                                               or (2² + 0² + 8² + 0²) / 4  -  2.5² = 10.75
       after 5 generations, when the number of Ton^r cells has doubled in each culture:
        variance =  [(5.0 - 4)² + (5.0 - 0)² + (5.0 - 16)² + (5.0 - 0)²] / 4 = 48.00

        Mutation Hypothesis predicts variance >> mean, as g increases

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All text material ©2026 by Steven M. Carr 

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