Events:
Q=allowed in the country
A=minor mistake
B=lie
C=missing documents
D=successful first appeal
E=failed the first appeal
G= successful second appeal
What you have so far:
Pr(Q)=Pr(D)+Pr(G∣E)
Pr(Q)=Pr(D∩A)+Pr(D∩B)+Pr(D∩C)+Pr(G∣E)
Pr(Q)=Pr(D∣A)Pr(A)+Pr(D∣B)Pr(B)+Pr(D∣C)Pr(C)+Pr(G∣E)
Note that:
1−Pr(D)=Pr(E)
What do you think about the probability in the blue?