The quality control manager of Bob's Cookies is inspecting a batch of chocolate chip cookies that has just been baked. If the production process is in control, the mean number of chip parts per cookie is 6.0. If the underlying distribution is assumed to be Poisson, what is the probability that in any particular cookie being inspected:
a. less than five chip parts will be found?
b. exactly five chip parts will be found?
c. five or more chip parts will be found?
d. either four or five chip parts will be found?
e. Graph the probability distribution for all possible number of chip parts per cookie.
P01 Bottleco produces six-packs of soda cans. Each can is supposed to contain at least 12 ounces of soda. If the total weight in a six-pack is under 72 ounces, Bottleco is fined $100 and receives no sales revenue for the sixpack. Each six-pack sells for $3.00. It costs Bottleco $0.02 per ounce of soda put in the cans. Bottleco can control the mean fill rate of its soda-filling machines. The amount put in each can by a machine is normally distributed with standard deviation 0.10 ounce. What mean fill quantity (within ±0.05 ounce) that maximizes expected profit per six-pack? P02 The Business School at State University currently has three parking lots, each containing 155 spaces. Two hundred faculty members have been assigned to each lot. On a peak day, an average of 70% of all lot 1 parking sticker holders show up, an average of 72% of all lot 2 parking sticker holders show up, and an average of 74% of all lot 3 parking sticker holders show up. Given the current situation, estimate the probability that on a peak day, at least one faculty member with a sticker will be unable to find a spot. Assume that the number of people who show up at each lot is independent of the number of people who show up at the other two lots. P03 A department store is trying to determine how many dresses to order for the spring season. Demand for the dresses is assumed to follow a normal distribution with mean 400 and standard deviation 100. The contract with the dressmaker works as follows. At the beginning of the season, the store reserves x units of capacity. The store must take delivery for at least 80% of the capacity it reserves and can, if desired, take delivery on up to all x dresses. Each dress sells for $160, and costs $50 per dress. If the store does not take delivery on all x dresses, it owes the dressmaker a $5 penalty for each unit of reserved capacity that was unused. For example, if the store orders 450 dresses, and demand is for 400 dresses, then the store will receive 400 dresses and owe the dressmaker 400($50) + 50($5). How many units of capacity should the store reserve to maximize its expected profit? P04 A martingale betting strategy works as follows. We begin with a certain amount of money and repeatedly play a game in which we have a 40% chance of winning any bet. In the first game, we bet $1. From then on, every time we win a bet, we get 2x the original bet and we bet $1 the next time. Each time we lose, we double our previous bet. Example: if we win the first bet, we get $2 and bet $1 for the next round If we lose the first bet, we lose the $1 and we bet $2 for the next round. If we win the $2 bet, we get $4 and reduce our next bet to $1 again. Currently we have $63. Assume we have unlimited credit, so that we can bet more money than we have. Use simulation to estimate the profit we will have earned after playing the game 50 times. Calcualte a 95% confidence interval for that profit. P05 The management of manufacturing company is considering the introduction of a new product. The fixed cost to begin the production of the product is $35,000. The variable cost for the product is constant for the amount to meet demand and is uniformly distributed between $16 and $24 per unit (remember, this cost should be modeled as dollars and cents, not just whole dollars). The product will sell for $50 per unit. Demand for the product is best described by a normal probability distribution with a mean of 1200 units and a standard deviation of 300 units. Run a Monte Carlo simulation with 1000 iterations. What is the mean profit you expect to achieve? What is the probability the project will result in a loss? P06 Suppose that Coke and Pepsi are fighting for the cola market. Each week, each person in the market buys one case of Coke or Pepsi. If the person's last purchase was Coke, there is a 0.90 probability that this person's next purchase will be Coke; otherwise, it will be Pepsi. (We are considering only two brands in the market.) Similarly, if the person's last purchase was Pepsi, there is a 0.80 probability that this person's next purchase will be Pepsi; otherwise, it will be Coke.Currently half of all people purchase Coke, and theother half purchase Pepsi. Simulate 1 year of sales in the cola market and estimate each company's average weekly market share. Do this by assuming that the total market size is fixed at 100 customers. P07 CallsProbability 800.10 1200.40 1600.30 2000.15 3000.05 A direct marketer of women's clothing, must determine how many telephone operators to schedule during each part of the day. They estimate that the number of phone calls received each hour of a typical 8-hour shift can be described by the probability distribution in the table. Each operator can handle 15 calls per hour and costs thecompany $20 per hour. Each phone call that is not handled is assumed to cost the company $6 in lost profit. Considering the options of employing 6, 8, 10, 12, 14, or 16 operators, use simulation to determine the number of operators that minimizes the expected hourly cost (labor costs plus lost profits). P08 Telephone calls arrive at the information desk of a large computer software company following an exponential distribution at a rate of 15 per hour. a. What is the probability that the next call will arrive within 3 minutes (0.05 hour)? b. What is the probability that the next call will arrive within 15 minutes (0.25 hour)? c. Suppose the company has just introduced an updated version of one of its software programs, and telephone calls are now arriving at a rate of 25 per hour. Given this information, redo (a) and (b ). P09 The quality control manager of Bob's Cookies is inspecting a batch of chocolate chip cookies that has just been baked. If the production process is in control, the mean number of chip parts per cookie is 6.0. If the underlying distribution is assumed to be Poisson, what is the probability that in any particular cookie being inspected: a. less than five chip parts will be found? b. exactly five chip parts will be found? c. five or more chip parts will be found? d. either four or five chip parts will be found? e. Graph the probability distribution for all possible number of chip parts per cookie.