20 GENETIC STOCKS AND BREEDING METHODS 



The probability of the undesired backcrosses in G n is t n and of the desired incrosses 



p tt = i-* B =i-(i -cy-\ 



If c = 1/2, the probability/? is successively 



0, 1/2, 3/4, 7/8, 15/16, 31/32,..., 



for n = 1, 2, 3, The probabilities for the desired mating types are shown in 



figure 4, for selected values of c . 



Fig. 4. Probability of incrosses for the backcross system. 

 1.0 



0.8 



0.6 



PROBABILITY 



OF Q4 



INCROSSES u - 



0.2 - 



0.0 



2 4 6 8 10 12 

 GENERATION^ 



The probability of incrosses p n for the backcross system starting with q = 1. The 

 curves shown are for five selected values of c = 1/10, 2/10, 3/10, 4/10, 5/10. The prob- 

 ability of heterozygosity is h n = 1 — p n . The ratio of successive values of h is constant 

 h n + i/K = I — c, when n 3s 1 . 



The probability h n of heterozygosity in G n , when n > 1 , is 

 K = r n + t n = (1 -f)"- 1 . 

 The ratio of successive values of h is constant 



or the probability of heterozygosity in one generation is the fraction 1 — c of the proba- 

 bility in the preceding generation. When c = 1/2, the probability of heterozygosity 

 is exactly cut in half with each additional generation of backcrossing. 



The generation matrix, G, for the backcross system is given in table 6 (a) . Because 

 of the fact, mentioned above, that only incrosses or backcrosses of the second type occur 

 after the initial generation, a simplified matrix, G*, can be used if G x is considered as 

 the initial generation. G* is given in table 6 (b) . Table 6 (c) , (d) , and (e) give the A, 

 A, and A~ l matrices for this system. 



