2. A group of 60 stocks includes 36 that have increased in value since November 1st, and 24 that have decreased since November 1st.
a) A.J.Jones has taken a random sample of 6 stocks from the group, and would like to calculate (before looking to see, of course) the probability of having picked at least 4 of the stocks that have increased in value. He can’t remember whether to use the binomial or hypergeometric formula. Help him out by demonstrating that for three decimal places of accuracy he can use either one.
b) If the stocks are examined in a random manner one-by-one, what is the probability that the first three will all be stocks that have increased?
c) If the stocks are examined in a random manner one-by-one, Let X be the number of stocks that have been examined when the first decreased-value stock is found. Find the median value of X.
4. a) Banks only accept coins for deposit if they are rolled in
specified amounts. Pennies, for example, are to be in rolls of 50. A Canadian bank has checked the 200 rolls of pennies they have on hand. Only 12 of the rolls contained all Canadian pennies: the rest all had at least one American penny. Altogether in the 200 rolls, about how many American pennies are there?
b) What is the median number of American pennies in a roll?
c) N.B. This c) part has nothing to do with the a) part.
The number of cars passing a checkpoint is treated by a consulting firm as having a Poisson distribution with a mean of 20 per minute. What does the firm consider the standard deviation in interarrival times to be?
5. a) In an office employees have been categorized according to
seniority and age. Some of the results are cross-tabulated below.
|Aged 50 or more||8||32|
|Aged less than 50||12||28|
i) If 10 workers from the office are selected at random,
what is the probability of at least 3 white-collar workers amongst those chosen?
ii) If 10 workers aged less than 50 are selected at random from the office, what is the probability of at most 3 management workers amongst those chosen?
b) A thick book with 1200 pages has quite a few typographical
errors. There are only 180 pages without typographical errors in the whole book. If typographical errors occur randomly, about how many pages in the book have three typographical errors?
d) What is the median number of typographical errors per page?
6. The length of time taken to learn a set of instructions may be treated as having an exponential distribution with a mean of 24 minutes. There is only one copy of the instructions, so as soon as one person has completed the learning process, the next person begins, with no intervening gap.
a) What proportion of those learning take between 20 and 30 minutes?
b) What is the 60th percentile of learning times?
c) What is the probability that at least three persons complete the learning process in a single one-hour period?
8. A.J.Jones works at Acme Economics Think Tank. The employees’ cafeteria offers a daily special called Box Lunch Surprise for $6.50. The cook has a limited repertoire, so that the surprise lunch is always either a ploughman’s lunch (cheese and bread) or a grilled-cheese sandwich with fries. On December 19th, 2011 there are 20 unmarked box lunches, of which there are 8 ploughman’s lunches and 12 grilled-cheese sandwiches. The boxes are arranged in a random fashion so that there is no way of knowing what is in a box before it is bought. Once opened, a box lunch cannot be returned.
a) Today A.J.Jones has decided to buy a Box Lunch Surprise for each of the 4 members of his team. What is the probability that there will be two ploughman’s lunches and two grilled-cheese sandwiches?
b) Two members of the team will absolutely not eat grilled cheese. How many lunches will A.J. have to buy in order to have at least a 90% probability of including two ploughman’s lunches (or more)?
c) A.J. buys boxes one at a time until he gets three grilled cheese. What is the probability that he will have to spend $45.50 in order to achieve his goal?
9. A time-and-motion-study consultant has been hired at Eurelia Industries Limited. She has identified a certain work station as a bottleneck in production. Initial data suggest that the times required for processing pieces at this station may be treated as having an exponential distribution with a mean of three hundred seconds.
a) If fifty pieces are processed, about how many of them take between two hundred forty and three hundred sixty seconds to process?
b) What is the probability that more than the mean number of pieces will be processed in a ten-minute period?
c) What is the median processing time at the station?
d) What proportion of processing times are within two standard deviations of the mean?