The BigVision Electronic Store sells a large 73 inch diagonal big screen TV. The TV comes with a standard 1 year warranty on parts and labor so that if anything malfunctions on the TV in the first year of ownership, the company repairs or replaces the TV for free. The store also sells an "extended warranty," which a customer can purchase that extends warranty coverage on the TV for another 2 years, for a total of three years of coverage. The daily numbers of TVs and extended warranties sold, and their probabilities, are represented by the following probability space \(\{S, \Upsilon, P\}\), where \(x\) denotes the number of TVs sold and \(y\) denotes the number of extended \(\Upsilon\) warranties sold:
\[S=\{(x, y): x \text { and } y \in\{0,1,2,3,4\}, x \geq y\}, \Upsilon=\{A: A \subset S\}\]
and \[P(A)=\sum_{(x, y) \in A} f(x, y)\]
where the nonzero values of \(f(x, y)\) are defined in the following table:

a. What is the probability that all of the TVs sold on a given day will be sold with extended warranties?

b. Given that \(\leq 2\) TVs are sold, what is the probability that all of the TVs will be sold with extended warranties?

c. Let A be the event that no TVs are sold, and B be the event that no extended warranties are sold. Are A and \(B\) independent events?

d. What is the probability that \(\geq 3\) extended warranties are sold on a given day?

e. What is the probability that more extended warranties are sold than TVs?

f. Find an algebraic representation of the function \(f(x, y)\) that can be used to replace the table of values above.

Number of TVs Number of extended warranties 0 1 2 3 0 1 NM + 2 3 4 0 4 0 0 0 0 .02 .03 0 .04 .05 .06 0 0 .06 .07 .08 .09 0 -08 .09 .10 .11 12 0 0