Say whether each of the following expressions is correct for any non-empty events A, B, and C.

a. P(A | B) =

P(A ∩ B)

P(A)

.

b. P(A | B) =

P(A ∩ B)

P(B)

.

c. P(A | B) + P(A | B) = 1.

d. P(A | B) + P(A | B) = 1.

e. P(A ∪ B | C) = P(A | C) + P(B | C) − P(A ∩ B | C).

f. P(A ∩ B) = P(A | B)P(B).

g. P(A | B) =

P(B | A)P(A)

P(B)

.

h. P(A ∩ B ∩ C) = P(A | B ∩ C)P(B ∩ C).
i. P(A | B ∪ C) = P(A ∩ B)
P(B ∪ C)
.
j. P(B | A)P(A) = P(A ∩ B).