Conditional Probability

  (1-r)/2 AQ r/2 Aq r/2 aQ (1-r)/2aq
(1-r)/2 AQ (1-r)2/4 r(1-r)/4 r(1-r)/4 (1-r)2/4
r/2 Aq r(1-r)/4 r2/4 r2/4 r(1-r)/4
r/2 aQ r(1-r)/4 r2/4 r2/4 r(1-r)/4
(1-r)/2 aq (1-r)2/4 r(1-r)/4 r(1-r)/4 (1-r)2/4

  QQ Qq qq  
AA (1-r)2/4 r(1-r)/4 + r(1-r)/4 r2/4 1/4
Aa r(1-r)/4 + r(1-r)/4 (1-r)2/4 + r2/4 + r2/4 + (1-r)2/4 r(1-r)/4 + r(1-r)/4 1/2
aa r2/4 r(1-r)/4 + r(1-r)/4 (1-r)2/4 1/4
  1/4 1/2 1/4 1.0

Prob.(QQ/AA) = Prob(QQ and AA) = (1-r)2/4 = (1-r)2
Prob(AA) 4

Prob.(Qq/Aa) = Prob(Qq and Aa)
Prob(Aa)

Prob.(Qq/Aa) = [(1-r)2/2 + r2/2] = (1-r)2 + r2
1/2

Marker
Genotype 		P(Qj/M)
QQ Qq qq
AA (1-r)2 2r(1-r) r2
Aa r(1-r) (1-r)2 + r2 r(1-r)
aa r2 2r(1-r) (1-r)2

Suppose we select individuals that have the AA marker genotype. What is the probability that the QTL genotype will be QQ?

P(Qj/M) = P(QQ/AA) = (1-r)2
The proportion of QQ selected.

This result shows that as r increases for coupling linkage between A and Q, selection for QQ will be less effective.


Let us select only individuals with the AA marker genotype. Now we have reduced the sample space.

  QQ Qq qq  
AA (1-r)2/4 r(1-r)/2 r2/4 1/4
Aa r(1-r)/2 (1-r)2/4 + r2/4 r(1-r)/2 1/2
aa r2/4 r(1-r)/2 (1-r)2/4 1/4


The reduced sample space is now:

  QQ Qq qq  
AA (1-r)2 2r(1-r) r2 1.0

P(QQ/AA) + P(Qq/AA) + P(qq/AA)
= (1-2r+r2) + (2r-2r2) + r2 = 1