1. A black velvet bag contains three red balls and three green balls. Each experiment involves drawing two balls at once, simultaneously, and recording their colors, R for red, and G for green.

(i) Obtain the sample space, assuming that balls of the same color are indistinguishable.

(ii) Upon assigning equal probability to each element in the sample space, determine the probability of drawing two balls of different colors.

(iii) If the balls are distinguishable and numbered from 1 to 6, and if the two balls are drawn sequentially, not simultaneously, now obtain the sample space and from this determine the probability of drawing two balls of different colors.

2. An experiment is performed by selecting a card from an ordinary deck of 52 playing cards. The outcome, ω, is the type of card chosen, classi?ed as: "Ace," "King," "Queen," "Jack," and "others." The random variable X(ω) assigns the number 4 to the outcome if ω is an "Ace;" X(ω) = 3 if the outcome is a "King"; X(ω) = 2 if the outcome is a "Queen," and X(ω) = 1 if the outcome  is  a "Jack"; X(ω) = 0 for all other outcomes.

(i) What is the space V of this random variable?

(ii) If the probability set function P (Γ) de?ned on the subsets of the original sample space Ω assigns a probability 1/52 to each of these outcomes, describe the induced probability set function PX (A) induced on all the subsets of the space V by this random variable.

(iii) Describe a physical (scienti?c or engineering) problem for which the above would be a good surrogate "model."