A) What is P(F)?

Using the law of total probability, we have:

P(F) = P(F|E)P(E) + P(F|E')P(E')

where E' is the complement of E. Since E and E' are mutually exclusive and exhaustive, we can write:

P(E') = 1 - P(E) = 1 - 0.3 = 0.7

Also, we are not given any information about P(F|E'), so we cannot compute it. Therefore:

P(F) = P(F|E)P(E) + P(F|E')P(E')
= 0.2(0.3) + P(F|E')0.7

b) What is P(G)?

Using the law of total probability again, we have:

P(G) = P(G|E)P(E) + P(G|E')P(E')

Since we are not given any information about P(G|E') either, we cannot compute it. Therefore:

P(G) = P(G|E)P(E) + P(G|E')P(E')
= 0.1(0.3) + P(G|E')0.7

c) What is P(F and G)?

Using the definition of conditional probability, we have:

P(F and G|E) = P(F|E)P(G|E)

Therefore, we can write:

P(F and G) = P(F and G|E)P(E) + P(F and G|E')P(E')

Again, we are not given any information about P(F and G|E'), so we cannot compute it. Therefore:

P(F and G) = P(F and G|E)P(E) + P(F and G|E')P(E')
= (0.2)(0.1)(0.3) + P(F and G|E')0.7

d) Are F and G independent?

We can check whether F and G are independent by comparing their joint probability P(F and G) with their product of individual probabilities P(F)P(G). If they are equal, then F and G are independent; otherwise, they are dependent.

From part c), we have:

P(F and G) = (0.2)(0.1)(0.3) + P(F and G|E')0.7

From parts a) and b), we have:

P(F) = 0.2(0.3) + P(F|E')0.7
P(G) = 0.1(0.3) + P(G|E')0.7

Therefore:

P(F)P(G) = [0.2(0.3) + P(F|E')0.7][0.1(0.3) + P(G|E')0.7]

Expanding and simplifying, we get:

P(F)P(G) = 0.06 + 0.02P(F|E') + 0.03P(G|E') + 0.49P(F|E')P(G|E')

Since we are not given any information about P(F and G|E'), we cannot compute it directly. However, we can use the formula for conditional probability:

P(F and G|E