These individual probabilities are given by the hypergeometric distribution and the result reduces to:where C is independent of N, and bl+bol+bool+bml = total number (D) of different antibody molecules detected in the whole sample.From this result, relative probabilities for different values of N can be calculated using Stirling's approximation for factorial For hand calculation, the value of N giving the greatest probability (i.e. the maximum likelihood estimate of N) can most easily be found by considering the ratio of these probabilities for successive values of N. The estimate is then the largest valur of N for which Probabilities corresponding t o the data of Table 4 (i.e. n l =138, n,=6l, n3=67, n4=71, D=332) are shown in Fig. 4. If all assumptions are correct, a value of approximately 8000 can be given as an estimate of N, although clearly there is even then a wide range of likely values.Within a particular donor/recipients system some antibody types may have a greater chance of being sampled than others. However assumption (ii) is invalid only if this difference depends on some property of the antibody types (i.e. if some types have a relatively high probability of selection in all donor/recipients systems). If assumption (ii) is invalid, or if some of the spectra that were considered identical were in fact produced by different antibodies, the above method might be expected t o underestimate N. We might then consider 2000 to represent the smallest likely value, but no upper limit could then be given.However, let us now consider the case in which each mouse is genetically capable of producing only A (less than N) different antibody types (i.e. assumption (i) is invalid), and the A different types are not necessarily the same for all four mice. If we accept assumption (ii), the method given above should in this case be regarded as providing an overestimate of A rather than an estimate of N. If A is only slightly less than N, the degree by which A is overestimated should be small and 2000 as a minimum estimate of both A and N is unlikely t o be misleading. If A is much less than N, then, in general, no meaning can be attached t o the estimate. However, in this particular case, estimates of the number of different antibodies actually produced by a donor mouse (based on an analysis of the frequency of occurrence of identical antibodies in different mice receiving cell transfer from the same donor mouse) suggest that 2000, as a minimum estimate of N, is still unlikely t o be misleading.