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A="b is in the middle"
B="c is to the right of b"
C="The letter def occur together in that order"
a) b can be in 7 places, but only one is the middle. So, P(A)=1/7
b) X=i, "b is in the i-th position"
Y=j, "c is in the j-th position"
P(B)=1/2
c) X=i, "d is in the i-th position"
Y=j, "e is in the j-th position"
Z=k, "f is in the i-th position"
P(C)=1/42
P(A∩C)=2*(1/7*1/6*1/5*1/4)=1/420
P(B∩A)=3*(1/7*1/6)=1/14
P(A|C)=P(A∩C)/P(C)=(1/420)/(1/42)=1/10
P(B|C)=P(B∩C)/P(C)=(1/420)/(1/42)=1/10
P(A|B)=P(B∩A)/P(B)=(1/14)/(1/2)=1/7
P(A∩B)=1/14
P(A)P(B)=(1/7)*(1/2)=1/14
A and B are independent
P(A∩C)=1/420
P(A)P(C)=(1/7)*(1/42)=1/294
A and C aren't independent
P(B∩C)=1/420
P(B)P(C)=(1/2)*(1/42)=1/84
B and C aren't independent