2. Winning Color In a population, the proportion of red elements is p,, the proportion of blue elements is pb , the proportion of green elements is pg, and p, + pb + p8 = 1. In each part below, set x equal to the quantity that you are trying to find, and develop an equation for x by conditioning on the first couple of draws. Try to write the simplest equation you can. Then solve for x. Do not use any other method. The point of this exercise is for you to learn how to use the method outlined above. It's almost certainly going to be shorter than any other correct method. 3) Alan draws repeatedly at random with replacement from the population, betting that the color red will appear before the color blue. Find the chance that he wins his bet. Your final answer should be in terms of p, and pb only, and you should aim for the simplest possible form. b) The answer to a can be applied in any situation where there are i.i.d. multinomial trials. For example, suppose a die is rolled repeatedly. Use your answer to Part a to find the chance that the face with one spot appears before all the faces that have an even number of spots. c) Now suppose Alan plays the following game with Katherine. They draw alternately at random with replacement from the population, with Alan drawing first. Alan wins if he draws red before Katherine has drawn blue. Katherine wins if she draws blue before Alan has drawn red. Find the chance that Alan wins the game. To keep your expressions simple, use the notation q, = 1 p, and qb = 1 pb. d) Find the expected duration of the game in Part c. That is, find the expected number of draws till there is a winner