Roll a fair six-sided die three times. Let A1 = {1 or 2 on the first roll}, A2 ={3 or 4 on the second roll}, A3 = {5 or 6 on the third roll}. It is given that P (Ai) = 1/3,i = 1, 2, 3; P (Ai ∩ Aj ) = (1/3)2, i̸ = j; and P (A1 ∩ A2 ∩ A3) = (1/3)3.(a) Use Theorem 1.1-6 (from the textbook) to find P (A1 ∪ A2 ∪ A3)(b) Given that P (A′1 ∩ A′2 ∩ A′3) = P (A′1)P (A′2)P (A′3) (which is due to independence,a concept that will be covered later), show that P (A1 ∪ A2 ∪ A3) = 1 − (1 − 1/3)3.

1. 1.1-6. If P (A) = 0.4, P (B) = 0.5, and P (A ∩ B) = 0.3, find(a) P (A ∩ B)(b) P (A ∪ B′)(c) P (A′ ∪ B′)2. 1.1-9 Roll a fair six-sided die three times. Let A1 = {1 or 2 on the first roll}, A2 ={3 or 4 on the second roll}, A3 = {5 or 6 on the third roll}. It is given that P (Ai) = 1/3,i = 1, 2, 3; P (Ai ∪ Aj ) = (1/3)2, i̸ = j; and P (A1 ∪ A2 ∪ A3) = (1/3)3.(a) Use Theorem 1.1-6 (from the textbook) to find P (A1 ∪ A2 ∪ A3)(b) Given that P (A′1 ∩ A′2 ∩ A′3) = P (A′1)P (A′2)P (A′3) (which is due to independence,a concept that will be covered later), show that P (A1 ∪ A2 ∪ A3) = 1 − (1 − 1/3)3.3. 1.1-10. Prove Theorem 1.1-61

3 pointsA fair die is rolled twice and a fair coin is tossed twice. Define eventsA : A three appear on the die twice.B : A head appear on the coin twice.Find the value of P(A ∩ B).1/61/721/121/144

fair die is rolled twice. Let 𝐴 be the event that an even number is obtained on the first rolland let 𝐵 be the event that a 3 is obtained on the second roll.(i) Find 𝑃(𝐴 ∩ 𝐵).(ii) Hence, or otherwise, find 𝑃(𝐴 ∪ 𝐵)

Question 2 of 10You roll a 6-sided die with faces numbered 1 through 6, and toss a coin. What is the probability of rolling a 3 or getting heads?A.B.C.D.

A six-sided die is rolled ( 10 ) (10) times, and the results are listed below. 3 3 , 4 4 , 1 1 , 1 1 , 2 2 , 6 6 , 6 6 , 5 5 , 6 6 , 3 3 What is the experimental probability of rolling a ( 3 ) (3) ?