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

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

A coin is flipped and a die is rolled. Find the probability of getting a tail on the coin and a 3 on the die.Question 3Answera.1/2b.1/12c.1/6d.8/12

A coin is tossed three times.Find the probability that at least 2 heads show up.

A fair coin is tossed, and a fair die is thrown. Write down sample spaces for(a) the toss of the coin;(b) the throw of the die;(c) the combination of these experiments.Let A be the event that a head is tossed, and B be the event that an odd number isthrown. Directly from the sample space, calculate P (A ∪ B) and P (A ∩ B)

ProbabilityTwo coins are tossed, find the probability that one head only is obtained.Options1/41/61/21/3