1. The first part of the practice is about the concept for probability and probability rules. Here are some key concepts as listed: Sample Space: All the possible outcomes for your observation from a random process. Event: A subset of the sample space. Probability: How likely an event would occur from the sample space. Probability rules: 0 For any event A, O s P(A) s 1 o For a sample space (S), P(S) = 1 0 Addition Rule: If two events A and B are disjoint, P(A or B) = P(A) + P(B) o Complement Rule: P(not A) = 1 P(A) o Multiplication Rule: If two events A and B are independent, P(A and B) = P(A)*P(B). (1) A fair die has 6 faces, and one face is colored red, two faces are colored blue, and rest three faces are colored green. If the die is rolled once, and its up-face COLOR is recorded. What is the sample space? What is the probability of each of the outcomes in the sample space? (Please verify that the total probability for the entire sample space is 1) (2) Now let's get it a bit more complicated. If we have two dice like the one above, and we will roll both dice, then record the up-face colors of each die. What is the sample space? What is the probability of each of the outcomes in the sample space? (Please verify that the total probability for the entire sample space is 1) Hint: One of the possible outcomes from the sample space could be {first die is Red and second die is Red}, which we can note as (Red, Red). Since {first die is Red} and {second die is Red} are independent from each other, we can use multiplication rule to get the probability of P(Red, Red) = P(first die is Red)*P(second die is Red) = (1/6)*(1/6) = 1/36. Color combinations of two dice robability ( 1/36 (3) We will make another change here. Instead of recording the color combinations, we will record the number of dices that has red up-face of the two dice we roll. What is the sample space? What is the probability of each outcome from the sample space? (Please verify that the total probability for the entire sample space is 1) Number of Red dice Probability