The possible outcomes for the sum of 9 when the die is thrown twice are: (3, 6), (6, 3), (4, 5), and (5, 4).

The probability of rolling a 3 is 0.1 and the probability of rolling a 6 is 0.3. Thus, the probability of (3, 6) is:

\(P(3, 6) = 0.1 \times 0.3 = 0.03\)

The probability of (6, 3) is the same as (3, 6) because the events are independent:

\(P(6, 3) = 0.3 \times 0.1 = 0.03\)

The probability of rolling a 4 is 0.2 and the probability of rolling a 5 is 0.1. Thus, the probability of (4, 5) is:

\(P(4, 5) = 0.2 \times 0.1 = 0.02\)

The probability of (5, 4) is the same as (4, 5):

\(P(5, 4) = 0.1 \times 0.2 = 0.02\)

Therefore, the total probability that the sum is 9 is:

\(P(\text{sum is 9}) = P(3, 6) + P(6, 3) + P(4, 5) + P(5, 4) = 0.03 + 0.03 + 0.02 + 0.02 = 0.1\)