Please help me solving the following problem:
At his workplace, the first thing Oscar does every morning is to go to the supply room and pick up one, two or three pens with probabilities 0.03, 0.23, 0.74 respectively. 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 probabilities 0.03, 0.23, 0.74 respectively. (The number of pens taken in one trip will not affect the number of pens taken in any other trip.)
Calculate the following precise to 4 digits:
- The probability that Oscar gets a total of three pens on any particular day.
- 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.
- 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}.
E[N]=- E[N|C]=
- σ2N|C, the conditional variance of the total number of pens Oscar gets on a particular day, where N and C are as in part (c).
- The probability that he gets more than three pens on each of the next 4 days.
- The conditional variance of the total number of pens he gets in the next 4 days given that he gets more than three pens on each of those days.