A="b is situated in the center"
B="c lies to the right of b"
C="The letters def occur sequentially in that arrangement"
a) b can occupy 7 positions; however, only one of these is the center. Therefore, P(A)=1/7
b) Let X=i; "b holds the i-th position"
Y=j; "c occupies the j-th position"
P(B)=1/2
c) Let X=i; "d holds the i-th position"
Y=j; "e occupies the j-th position"
Let 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
Events A and B are independent
P(A∩C)=1/420
P(A)P(C)=(1/7)*(1/42)=1/294
Events A and C are not independent
P(B∩C)=1/420
P(B)P(C)=(1/2)*(1/42)=1/84
Events B and C are not independent