In Equation (2.33), as n approaches N, ny in the numerator will cancel out with Ny in the denominator. Also, after the cancellation, all the factors in the numerator of the second factor will approach 1. Furthermore, as N approaches infinity, (1 - X/N)-y approaches 1. Lastly, as n approaches infinity, we have from calculus limitn^ (1 - X/N)n = ex. Applying all these to Equation (2.33) produces y x_ .
This distribution is called the Poisson distribution. This will be used to derive the method of estimating the most probable number, MPN.
Note the difference between the Binomial distribution and the Poisson distribution. In the former case, the sample is of fixed size, n, and the probability of obtaining y number of bacteria from this sample size is computed. In the Poisson distribution, since n is made to approach infinity, the sample size is the population, itself. In other words, although the sample is of fixed size, the probability is calculated as if the size is that of the population. This is fortunate, since it is impossible to compute the probability using the Binomial distribution; the sample size n is practically not known, unless the sample is painstakingly analyzed to count the number of organisms represented by the value of n, thus the use of the Poisson distribution.
2.3.4 Estimation of Coliform Densities by the MPN Method
As the phrase "most probable number" implies, whatever value is obtained is just "most probable." It also follows that the techniques of probability will be used to obtain this number.
Consider the 10-mL portions (dilution) inoculated into the five replicate tubes. If a is the expected number of bacteria per mL, the expected number of bacteria in the 10 mL (in one tube) is X = 10a. Thus, from the Poisson probability, for any one tube, the probability that there will be no bacteria (X = y = 0) is n i / t^ rw X -x (10a) -10a -10a
Let q be the number of negative tubes and p be number of positive tubes in the five replicate tubes. (Note that the symbols q and p hold for any number of replicate tubes, not only 5 tubes.) These q negative results occur at the same time, so they
The p positive tubes also are intersection events. If the probability of a negative result is e~10a, the probability of a positive result is 1 - e~10a. By analogy, the intersection probability of the p events is
Of course, the q negative results and the p positive results are all occurring at the same time; they are also intersection events to each other. Combining the q and the p results constitutes the dilution experiment. Because the intersection of the q and the p events is the dilution, call the corresponding probability as Prob(D), where D stands for dilution. Thus, from the intersection probabilities of the q and p events, the probability of the dilution is
Let the number of dilutions be equal to j number, where each dilution has a sample size of m mL (instead of the 10 mL). Each of these j dilutions will also be happening at the same time, and are, therefore, intersection events among themselves. Thus, in this situation, Equation (2.38) becomes i=j
where n is the symbol for the product factors. The product factors are used, because the j number of dilutions are intersection events.
Now, we finally arrive at what specifically is the quantitative meaning of MPN; this is gleaned from Equation (2.39). In this equation, for a given number of serial dilutions and respective values of q and p, the value of a that will make Prob(D) a maximum is the most probable number of organisms. Thus, to get the MPN, the above equation may be differentiated with respect to a and the result equated to zero to solve for a. This scheme is, however, a formidable task. The easier way would be to program the equation in a computer. For a given number of serial dilutions and corresponding values of q and p, several values of a are inputted into the program. This will generate corresponding values of Prob(D), along with the inputted values of a. The largest of these probability values then gives the value of a that represents the MPN. A table may then be prepared showing serial dilutions and the corresponding MPN.
The American Public Health Association has generated a statistical table for the MPN (APHA, AWWA, WEF, 1992). A modified version is shown in Table 2.4. The first three columns are the combination of positives in only three serial dilutions. Thus,
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