Dice probability problems

1 A biased die has faces numbered 1 to 6 . The probabilities of the die landing on 1,3 or 5 are each equal to 0.1 . The probabilities of the die landing on 2 or 4 are each equal to 0.2 . The die is thrown twice. Find the probability that the sum of the numbers it lands on is 9 .

1. A fair die has six faces numbered \(1,2,2,3,3\) and 3 . The die is rolled twice and the number showing on the uppermost face is recorded each time.

Find the probability that the sum of the two numbers recorded is at least 5 .
(5)

Two fair twelve-sided dice with sides marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 are thrown, and the numbers on the sides which land face down are noted. Events \(Q\) and \(R\) are defined as follows. \(Q\): the product of the two numbers is 24. \(R\): both of the numbers are greater than 8.

1. Find \(\mathrm{P}(Q)\). [2]
2. Find \(\mathrm{P}(R)\). [2]
3. Are events \(Q\) and \(R\) exclusive? Justify your answer. [2]
4. Are events \(Q\) and \(R\) independent? Justify your answer. [2]