FALL 2016/17 TERM 1 STAT 302 ASSIGNMENT 2 Due: Wednesday October 12 at 11am Please remember to include a cover sheet when you submit your assignment. You may hand in your assignment in class or deposit it in the STAT 302 assignment box on the ground floor of the Earth Sciences Building (ESB) by the due time. When answering the questions, writing down the final answer will not be sufficient to receive full marks. Please show all calculations unless otherwise specified. Also define any events and notation that you use in your solutions. 1. You work for the RCMP and are in charge of a search for a group of lost hikers. You have partitioned the search area into 3 sections of equal size, and the hikers on day 1 of the search have equal probability of being in any of the three sections. Your team is able to search only one of the sections per day. The search area is partitioned as follows: 1 2 3 Section 1 is a dense forest. If your team searches section 1 and if the hikers are in section 1, there is a 0.35 probability that your team will not find them. Section 2 is a flat grassland. If your team searches section 2 and if the hikers are in section 2, there is a 0.1 probability that your team will not find them. Section 3 is mountainous. If your team searches section 3 and if the hikers are in section 3, there is a 0.2 probability that your team will not find them. (a) Assume that the hikers do not move between sections. Suppose, on day 1, your team searches section 1 and is unable to find the hikers. For each of the three sections, find the probability that the hikers are in a given section after the search on day 1 is concluded. (b) It is possible that the hikers moved sections overnight. If the hikers were in section 1, they could move to section 2 with probability 0.05. If the hikers were in section 2, they could move to either section 1 or 3, each with probability 0.05. If the hikers were in section 3, they could move to section 2 with probability 0.05. All other movements have probability 0. Find the updated probabilities of the hikers' location at the start of day 2. Which section would you search on day 2, if you were to pick the section with the highest probability? 1 2. We would like to use a balance to weigh an item (the balance weights are allowed to be used on only one disk). We have three different sets of balance weights (in grams): (1) 1, 2, 2, 5, 10; (2) 1, 2, 3, 4, 10; (3) 1, 1, 2, 5, 10. When we measure the weight of an item, only one set of balance weights is allowed to be used. Assume that an item can only weigh 1g, 2g, 3g, ..., 10g, all with equal probability. Let Xi denote the number of balance weights we need to weigh an item when using the ith set, where i = 1, 2, 3. Which set of balance weights has the smallest expected number of weights used to weigh an item? 3. A retail store owner has overstocked a certain item and decides to use a promotion on this item in order to increase the sales. This item has a marked price of $100. The rules for this promotion are as follows. For every customer purchasing this item, the owner lets the customer roll a fair 6-sided die. If the customer rolls a \"1\" or \"6\