Finding minimum stock level for demand

Questions requiring the minimum number of items to stock so that the probability of meeting demand exceeds a threshold (practical inventory problems).

6 questions

CAIE S2 2016 June Q6
6 At a certain shop the demand for hair dryers has a Poisson distribution with mean 3.4 per week.
  1. Find the probability that, in a randomly chosen two-week period, the demand is for exactly 5 hair dryers.
  2. At the beginning of a week the shop has a certain number of hair dryers for sale. Find the probability that the shop has enough hair dryers to satisfy the demand for the week if
    (a) they have 4 hair dryers in the shop,
    (b) they have 5 hair dryers in the shop.
  3. Find the smallest number of hair dryers that the shop needs to have at the beginning of a week so that the probability of being able to satisfy the demand that week is at least 0.9 .
CAIE S2 2005 November Q6
6 A shopkeeper sells electric fans. The demand for fans follows a Poisson distribution with mean 3.2 per week.
  1. Find the probability that the demand is exactly 2 fans in any one week.
  2. The shopkeeper has 4 fans in his shop at the beginning of a week. Find the probability that this will not be enough to satisfy the demand for fans in that week.
  3. Given instead that he has \(n\) fans in his shop at the beginning of a week, find, by trial and error, the least value of \(n\) for which the probability of his not being able to satisfy the demand for fans in that week is less than 0.05 .
Edexcel S2 2017 October Q3
3. In a shop, the weekly demand for Birdscope cameras is modelled by a Poisson distribution with mean 8 The shop has 9 Birdscope cameras in stock at the start of each week. A week is selected at random.
  1. Find the probability that the demand for Birdscope cameras cannot be met in this particular week. In a year, there are 50 weeks in which Birdscope cameras can be sold.
  2. Find the expected number of weeks in the year that the shop will not be able to meet the demand for Birdscope cameras.
  3. Find the number of Birdscope cameras the shop should stock at the beginning of each week if it wants the estimated number of weeks in the year in which demand cannot be met to be less than 2 The shop increases its stock and reduces the price of Birdscope cameras in order to increase demand. A random sample of 10 weeks is selected and it is found that, in the 10 weeks, a total of 95 Birdscope cameras were sold. Given that there were no weeks when the shop was unable to meet the demand for Birdscope cameras,
  4. use a suitable approximation to test whether or not the demand for Birdscope cameras has increased following the price reduction. You should state your hypotheses clearly and use a 5\% level of significance.
Edexcel S2 2013 June Q3
  1. An online shop sells a computer game at an average rate of 1 per day.
    1. Find the probability that the shop sells more than 10 games in a 7 day period.
    Once every 7 days the shop has games delivered before it opens.
  2. Find the least number of games the shop should have in stock immediately after a delivery so that the probability of running out of the game before the next delivery is less than 0.05 In an attempt to increase sales of the computer game, the price is reduced for six months. A random sample of 28 days is taken from these six months. In the sample of 28 days, 36 computer games are sold.
  3. Using a suitable approximation and a \(5 \%\) level of significance, test whether or not the average rate of sales per day has increased during these six months. State your hypotheses clearly.
Edexcel S2 Q2
2. The number of copies of The Statistician that a newsagent sells each week is modelled by a Poisson distribution. On average, he sells 1.5 copies per week.
  1. Find the probability that he sells no copies in a particular week.
  2. If he stocks 5 copies each week, find the probability he will not have enough copies to meet that week's demand.
  3. Find the minimum number of copies that he should stock in order to have at least a \(95 \%\) probability of being able to satisfy the week's demand.
Edexcel S2 Q4
4. A hardware store is open on six days each week. On average the store sells 8 of a particular make of electric drill each week. Find the probability that the store sells
  1. no more than 4 of the drills in a week,
  2. more than 2 of the drills in one day. The store receives one delivery of drills at the same time each week.
  3. Find the number of drills that need to be in stock after a delivery for there to be at most a 5\% chance of the store not having sufficient drills to meet demand before the next delivery.
    (3 marks)
    [0pt]