Tina Fey goes trick-or-treating the night of Halloween dressed as Sarah Palin. She visits homes in her neighborhood to collect candy, but only receives candy, naturally, when the door is answered and the family still has a piece of candy to give away. Upon knocking, the probability of the door being answered is 3/4, and the probability that the home still has candy is 2/3. Assume that the events “Door answered” and “candy remaining” are independent and also that the outcomes at each home are independent. Also assume that each home gives away at most a single piece of candy. (a) Determine the probability that Tina receives her first piece of candy at the third house she visits. (b) Given that she has received exactly four pieces of candy from the first eight houses, determine the conditional probability that Tina will receive her fifth piece of candy at the eleventh house. (c) Determine the probability that she receives her second piece of candy at the fifth house.

Question

Tina Fey goes trick-or-treating the night of Halloween dressed as Sarah Palin. She visits homes in her neighborhood to collect candy, but only receives candy, naturally, when the door is answered and the family still has a piece of candy to give away. Upon knocking, the probability of the door being answered is 3/4, and the probability that the home still has candy is 2/3. Assume that the events “Door answered” and “candy remaining” are independent and also that the outcomes at each home are independent. Also assume that each home gives away at most a single piece of candy.

(a) Determine the probability that Tina receives her first piece of candy at the third house she visits.

(b) Given that she has received exactly four pieces of candy from the first eight houses, determine the conditional probability that Tina will receive her fifth piece of candy at the eleventh house.

(c) Determine the probability that she receives her second piece of candy at the fifth house.

(d) Given that she did not receive her second piece of candy at the second house, determine the conditional probability that she will receive her second piece of candy at the fifth house.