Please review Example 3.6, Fuel for Thought. Now suppose that Mendel has three gas stations on his route from home to work, and that the price of gas at each station is equally likely to be anywhere from $3.70 to $4.00 per gallon, with the prices at different stations independent. His strategy for getting a low price is: If the price is $3.80 or below at the gas station nearest his home, he fills up his gas tank there. If the price exceeds $3.80, he goes on to the second station, and fills up his tank there if the price is $3.90 or below. Otherwise, he proceeds to the third station. If he goes on to the third station, he fills up his tank there whatever the price. (i) Show that, given that Mendel buys gas at the first station, the probability that he paid y or less than y is (y - 3.70)/(3.80 - 3.70) for y < 3.80. In other words, his price given that he bought at the first station is U(3.70, 3.80). (ii) For each of the three gas stations, what is the probability that Mendel purchased his gas there? (iii) For each of the three stations, what is the conditional mean price that Mendel paid given that he bought his gas there? (iv) What is the overall mean price at which Mendel purchased his gasoline? (v) Suppose that Mendel is willing to change his threshold price at the second gas station from $3.90. Give an example of a change in that threshold that would reduce his average purchase price. (vi) Assuming that his second threshold stays at $3.90, are there any changes in the first threshold of $3.80 that would reduce his average purchase price? (vii) If he is willing to change both his first and second thresholds from $3.80 and $3.90 respectively, what might you suggest he do to lower his average purchase price?