At his workplace, the ?rst thing Oscar does every morning is to go to the supply...

At his workplace, the ?rst thing Oscar does every morning is to go to the supply room and pick up one, two, or three pens with equal probability 1/3. If he picks up three pens, he does not return to the supply room again that day. If he picks up one or two pens, he will make one additional trip to the supply room, where he again will pick up one, two, or three pens with equal probability 1/3. (The number of pens taken in one trip will not a?ect the number of pens taken in any other trip.) Calculate the following:

(a) The probability that Oscar gets a total of three pens on any particular day.

(b) The conditional probability that he visited the supply room twice on a given day, given that it is a day in which he got a total of three pens.

(c) E[N] and E[N |C], where E[N] is the unconditional expectation of N, the total number of pens Oscar gets on any given day, and E[N |C] is the conditional expectation of N given the event C = {N > 3}.

(d) ?N |C, the conditional standard deviation of the total number of pens Oscar gets on a particular day, where N and C are as in part (c).

(e) The probability that he gets more than three pens on each of the next 16 days. (f) The conditional standard deviation of the total number of pens he gets in the next 16 days given that he gets more than three pens on each of those days.