I need detailed answers to the questions

1. Let be the set of all students who passed Calculus I, be the set of all students who passed calculus II, and be the set of all students who passed calculus III.
a. i) = Set of all students who passed Calculus I, Calculus II and Calculus III.
ii) = Set of students who passed Calculus III.
iii) = Set of students who passed Calculus I or Calculus III.
iv) = Set of students who passed Calculus I and Calculus II.
v) = empty set = {
b. i) 1-
    ii)
2.
.
. Therefore,

Now,

3.




4.

(
1/4
) (
E
) (
E
) (
F
) (
E
)
(
0
) (
1/4
) (
1/2
) (
G
)        
Since E, F and G are mutually exclusive events hence, we have Now, the axiom of total probability asserts that the sum of all probability =1. Here, we already have the sum equal to 1, i.e., This shows that all possible probability...