There are four cards in a pocket, and each of them is marked with 3 numbers. These numbers are 110, 101, 011, and 000. You draw a card randomly and let Ai (i = 1, 2, 3) represent the event that your card's ith digit is a 1. For example: A2 represents the event that the 2nd digit is a 1. (Please show your work.)
a) List the outcomes in A1. Find P(A1).
b) Find P(A1 U A2).
c) Find P(A2|A3).
d) Find P(A3 ? A1)
e) Is A1 independent from A2? (prove your answer mathematically)
f) Are A1, A2 and A3 pairwise independent? (prove your answer mathematically)
g) Are A1, A2 and A3 mutually independent? (prove your answer mathematically)