a,b,c please write neatly and all questions.

4. A company has come out with a new carbonated soda to compete against the leading brand in a particular soda category. The company is considering a marketing campaign where they would set up tasting tables at music festivals and invite attendees to do a blind taste test: each taster would undertake three trials where in each they would be presented with two unmarked cups (marked only with A & B) and after tasting each of the two cups they would state which they preferred (cup A or cup B). (Of course, there is random assignment of the new soda and the leading brand to cup A or cup B in each trial.) Aer the completion of the three trials, the soda brands would be revealed, and the individual would be told how many times (out of three trials) they preferred the new soda to the leading brand. I Assume each of the trials is independent (preferences between the two sodas don't change in successive tastings). - Assumes that the company believes that in a single trial, the probability that such a taster will choose the new soda is 0.60. o In any calculations below, feel free to round to three or four decimal places. A. Define the random variable X to be the number of times a taster says they prefer the new soda in three trials. What is the probability distribution of random variable X ? B. Calculate E[X] and Var[X]. C. Suppose company decision makers dene success by the proportion of individual tasters that choose the new soda in the majority of their trials; i.e., they say that they prefer the new soda in two or three of their three trials. Suppose that 1,000 individuals undertake the taste test described. Define a random variable S to be the number of the 1,000 individuals that prefer the new soda in the majority of their trials. What is E [S] and Var[S] ? Hint: Dene anewrandomvariable Z=l if X22 and Z=0 if X51