I'm just working on this past exam paper question and would like to know if I'm doing everything correctly. Here is the question;
An experiment consists of three fair, different coloured dice being rolled (the dice are 6-sided and the sides show numbers 1, . . . , 6). Let A be the event that none of the dice shows numbers 1 and 2, and let B be the event that all dice show an odd number.
(a) What is the probability of A?
(b) What is the probability of B?
(c) What is the probability of A ∩ B?
(d) Are the events A and B independent? (You must justify your answer!)
I have answered the question and would like your feedback;
a) P(A) = 4/6 = 2/3 So because there are three dice = 2/3 * 2/3 * 2/3 = 8/27
b) P(B) = 3/6 = 1/2 So because there are three dices = ½ * ½* ½ =1/8
c) P(A ∩ B) = 2/6 =1/3 So because there are three dices = 1/3 * 1/3* 1/3* = 1/27
d) The events are not independent as P(A and B) do not equal to P(A)P(B), because the outcomes of one, impact the other.
For part (d) Note that the set $\{1,2\}$ includes one odd and one even element. So, $P(A \vert B)=P(A)$. If you follow the definition $$P(A \cap B)=P(A \vert B) P(B)=P(A)P(B)$$