Please need proper solution

Question

Two dice are cast: 𝐸1is the event that a 6 appears on at least one die, 𝐸2 is the event that a 5 appears on exactly one die and 𝐸3 is the event that same number appears on both dice.

i. Are 𝐸1 and 𝐸2 independent?

ii. Are 𝐸2 and 𝐸3 independent?

iii. Are 𝐸3 and 𝐸1 independent?

Answer 1

P(E1) = 11/36 (P(6 on first die) + P(6 on second die) - P(6 on both die))

P(E2) = 10/36 = 5/18 (P(5 on first die and no 5 on second die) + P(5 on second die and no 5 on first die))

P(E3) = 1/6 (P(1 on both dice) + P(2 on both dice) + P(3 on both dice) + P(4 on both dice) + P(5 on both dice) + P(6 on both dice))

i) P(E1E2) = 1/18 (P(5 on first die and 6 on second die) + P(6 on first die and 5 on first die))

P(E1) * P(E2) = 11/36 * 5/18 = 55/648 is not equal to 1/18 not independent

ii) P(E2E3) = 0 (if 5 on only one die, not possible to have same number on both dice) P(E2) * P(E3) = 5/18 * 1/6 = 5/108 is not equal to 0 not independent

iii) P(E1E3) = 1/36 (P(6 on both dice)) P(E1) * P(E3) = 11/36 * 1/6 = 11/216 is not equal to 1/36 not independent