Each time a button is pressed on a particular machine, a random integer between 1 and 2N is displayed on a screen, where N > 1 is a certain positive integer. All numbers between 1 and 2N are equally likely and the same number can appear more than once. The button is pressed m times and the displayed numbers are recorded. Consider the following three events: A: The first number displayed is either 1 or 2. B: The product of the m displayed numbers is even. C: The sum of the m displayed numbers is m+2. i. Find the probabilities P(A), P(B) and P(C). ii. Are the events A and B independent? iii. Are the events B and C independent? You can use the fact that 2" > x for all x > 0