Practice section 2 of 8

Two events A and B satisfy $P(A)=1/4$, $P(A\mid B)=1/2$ and $P(B\mid A)=2/3$.

(i) Determine whether A and B are independent, justifying your answer.

(ii) Determine whether A and B are mutually exclusive, justifying your answer.

(iii) Find P(B).

Events A and B satisfy $P(B)=2/3$, $P(A\mid B)=3/5$ and $P(A\cap B')=1/4$. State with a reason whether A and B are independent.

A fair die is thrown twice. Find the probability that (i) the total score is 6, (ii) the first throw is even, (iii) the total score is 6 given that the first throw is even, (iv) the first throw is even given that the total score is 6.

Two ordinary dice are thrown. Find the probability that (i) their sum is 3, (ii) their sum exceeds 9, (iii) they show the same number, (iv) their numbers differ by more than 2, (v) their product is even.

In a certain sample space, A and B are independent, $P(A\cup B)=4/9$ and $P(A\cap B')=5/18$. Calculate (i) P(B), (ii) P(A), (iii) $P(A\cup B')$.

Given $P(A)=0.7$, $P(B)=0.5$ and $P(A\mid B)=0.8$, find (i) $P(A\cap B)$, (ii) $P(A\cup B)$, (iii) $P(B'\mid A)$.