Difficult Probability Question! (involving independent or dependent events) 
Each time a button is pressed on a particular machine a random integer between 1 and 2Y is displayed on a screen, where Y is a certain positive integer. All numbers between 1 and 2Y are equally likely and the same number can appear more than once. The button is pressed n times and the displayed numbers are recorded. Consider the following three events:
the first number displayed is either 1 or 2.
B: the product of the n displayed numbers is even.
C: the sum of the n displayed numbers is n + 2.

(a) Find the probabilities P(A), P(B) and P(C).
(b) Are the events A and B independent?
(c) Are the events B and C independent?
Would P(A) simply be (1/Y)?
For P(B) will it be 1- [(Y^m)/((2Y)^m)] ?
(Essentially the probability that the probability that product of the displayed numbers is odd which mean all n displayed numbers have to be odd so the chance of picking an odd number (1/2) to the power of n times. Then that probability taken away from 1 will give the chance that one or more of the n displayed numbers are even and therefore the product is even)
For P(C) will it be [(n!)/(2!)((n-2)!)][(1/2Y)^2][(1/2Y)^(n-2)] + [n][(1/2Y)]
[(1/2Y)^(n-1)]
(Essentially for the sum of the n displayed numbers to be n+2 you need (n-2) displayed numbers to be 1 and 2 displayed numbers to be 2 OR (n-1) displayed numbers to be 1 and 1 displayed number to be 3.) So I did NC2 multiplied by the probability of the number 2 being generated twice and then the number 1 being generated (n-2) times. And then plus NC1 multiplied by the probability of the number 3 being generated once multiplied by the number 1 being generated (n-1) times.
Are the probabilities P(A), P(B) and P(C) correct?
If so how do I calculate P(B∩C)for part(c)
Any help would be much appreciated!
