A student throws a dice twice, and get two numbers, respectively. Let A
be the event for the sum of the two numbers is AT LEAST 8; Bi
be the event that the
first number is i, for i=1 to 6.

(i) Give the values of P(A|B1), P(A|B2), P(A|B3), P(A|B4), P(A|B5), and P(A|B6)?
[Hint: P(A|B1) is the probability that the sum of the two numbers is at least 8,
under the condition that the first number is 1; and same for the others.]
(ii) Using (i), find P(A).
(iii) Using (i) and (ii), prove that for i=1, 2, …,6, A and Bi
are not independent.
(Need to answer for A and B1, A and B2, …, A and B6.)
(iv) Give the values of P(B1|A), P(B2|A), P(B3|A), P(B4|A), P(B5|A), and P(B6|A)?
[Hint: P(B1|A) is the probability that the first number is 1 under the condition
that the sum of the two numbers is at least 8; and similar to the others.]