A student rolls a die until the first "4" appears. Let X be the numbers of rolls required until (and including) this first "4." After this is completed, he begins rolling again until he gets a "3." Let Y be the number of rolls, after the first "4," up to (and including) the next "3." E.g., if the sequence of rolls is 213662341261613 then X = 8 and Y = 7.
a. Let Aj be the event containing all outcomes in which "j or more rolls" are required to get the first "4." Let Xj indicate whether Aj occurs. Then X = X1 + X2 + X3 + . . . . Find the expected value of X by finding the expected value of the sum of the indicator random variables.
b. Let Bj be the event containing all outcomes in which uj or more rolls" are required, after the first "4," until he gets a "3." Let Yj indicate whether Bj occurs. Then Y = Y1 + Y2 + Y3 + . . . . Find the expected value of Y by finding the expected value of the sum of the indicator random variables.