Most Probable Number Equations & References
Assumption
As explained in [1: Download the Article Here], there are two assumptions.
1. The liquid is completely mixed so that the organisms in the liquid are randomly distributed.
2. “Each sample from the liquid, when incubated in the culture medium, is certain to exhibit
growth whenever the sample contains one or more organisms.” Otherwise, the MPN
underestimates the true density of organisms.
Calculation
Consider the single organism. The probability that the organism is present in the sample is simply given
by the ratio of the volume of the sample v to the total volume V and the probability that the organism
is not in the sample is given by
1 v
V
Due to the completely mixed assumption, this probability is irrelevant to the locations of the other
organisms. Assume that there are b organisms in the liquid. The probability that the sample does not
contain any of the organisms, p s , is given by

ps  1  v
If v
V

V
b
is small, this probability is described by

p s  exp  vb
V
Since b
V

is the density  of organisms, the probability is given by
p s  exp  v 
Therefore, the probability of obtaining a fertile sample, p f , is
p f  1  exp  v 
If n samples of volume v are taken, the probability observing s fertile samples is given by the binomial
distribution as
n!
s
ns
p f 1  p f 
s!(n  s)!
n!
1  exp  v s exp  v ns

s!(n  s)!
f s |   
What we want to know is the “most probable number (MPN)”, the density which maximizes the
probability of obtaining the observed result. Such a density is obtained by finding root of the derivative
of log likelihood function.

n! 
  s log 1  exp  v   n  s  v 
l    log 
 s!(n  s )! 
l  
v exp( v )
s
 n  s v  0

1  exp  v 
Therefore, the MPN is given by the root of the following equation
sv
 nv
1  exp  v 
In general, multiple dilution sets are used to obtain the MPN. The joint probability of observing s i fertile
samples in one of m dilution sets with volume v i and ni samples is given by the following likelihood
function.
m
L    1  exp  v  i exp  v  i
s
n  si
i 1
Like a single dilution case, the MPN is given by the root of the following equation
m
m
si vi

ni vi


i 1 1  exp  vi  
i 1
The MPN obtained by the above method can be approximated by the log normal distribution with mean
equal to the logarithm of the MPN and standard error as shown in the following equation [2: Download
the Article Here]. References [3,4,5] provide further explanation on the derivation of these equations.
SElog
2
m


vi ni
2

  MPN 

i 1 exp vi MPN   1 

1
2
Therefore, 95% confidence bound is obtained by the following equation.
Upper Bound  exp log MPN   1.96  SElog 
Lower Bound  exp log MPN   1.96  SElog 
IDEXX QuantiTray 2000
IDEXX QuantiTray 2000 consists of 49 large wells of volume 1.86ml and 48 small wells of volume
0.186ml. Therefore the above equation is described as
s1 1.86
s2  0.186

 49 1.86  48  0.186
1  exp  1.86  MPN  1  exp  0.186  MPN 
where S1 and S2 is the number of positive large and small wells respectively.
References
[1] William G. Cochran, Estimation of bacterial densities by means of the ‘Most Probable Number’,
Biometrics 6 (1950), no. 2, 105–116.
[2] M. A. Hurley and M. E. Roscoe, Automated statistical-analysis of microbial enumeration by dilution
series, Journal of Applied Bacteriology 55 (1983), no. 1, 159–164.
[3] Fisher, R.A. On the mathematical foundations of theoretical statistics. Philosophical Transactions of
the Royal Society of London (1921), A 222, 309-368.
[4] Loyer, M.W. and M.A. Hamilton. Interval Estimation of the Density of Organisms Using a SerialDilution Experiment, Biometrics, Vol. 40, No. 4 (Dec., 1984), pp. 907-916
Other References
[5] Finney, D.J. 1964 Statistical Methods in Biological Assay, 2nd Ed. High Wycombe: Charles Griffin &
Co.