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)

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

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.

Let A𝐴 and B𝐵 are two events and P(A′)=0⋅3𝑃𝐴'=0·3, P(B)=0⋅4, P(A∩B′)=0⋅5𝑃𝐵=0·4, 𝑃𝐴∩𝐵'=0·5 then P(A∪B′)𝑃𝐴∪𝐵' is:

Below is a list of all possible outcomes in the experiment of rolling two die.Note: The table gives the values of each die, not the sum of the two die.Determine the following probabilities. Write your answers as reduced fractions.P𝑃 (sum is even) =  P𝑃 (sum is 3) =  P𝑃 (sum is 10) =

A fair 6-sided die is rolled three times. What is the probability of rolling an even number at least once?