1. A society has as members 20 women, 18 men, and 30 children.
a) How many ways are there to select a committee of 3 women, 2 men, and 1 child?
5232600
b) How many ways are there to select a committee of 6 members if the only restriction is that there be at least 1 woman on the committee?
97181832
c) How many ways are there to select a committee of 6 members if the only restriction is that there be at least 1 woman and at least 1 man on the committee?
81884907
2. The probability of success for a given experiment is 0.36 (every time)
a) If there are twelve trials, what is the probability that the number of successes will be within one standard deviation of the expected number of successes ?
E(X) = 4.32 σ(X) = 1.66... P(3) + P(4) + P(5) = 0.63
b) What is the probability that the fifth success will occur before the ninth trial ? (try lateral thinking)
0.118
3. A new configuration of the computer key board has 10 of the 26 letters
of the alphabet in different positions from the standard keyboard, and
the rest are in the usual places. If a perfectly accurate touch typist types the word computer on the new keyboard without looking, what is the probability that there will be at least two mistakes in the resulting printed word ? It may be assumed that the typist does not know the new configuration.
Hypergeometric 0.9185
4. The probability of success for a given experiment is 0.6 (every time)
a) If there are twelve trials, what is the probability that there will be more than the mean number of successes ?
0.438
b) What is the probability that the fifth success will occur on either the tenth or eleventh trial ? (hard)
0.1672
5. In a group of 1875 scientists, 900 were educated in Canada, and the rest were educated elsewhere. Only 75 of the scientists were educated in Eurelia. A random selection was made of eight of the scientists.
a) What is the exact probability that at most four of them were educated in Canada ? Use an approximate distribution to get an approximation to the exact answer as well.
Exact (hypergeometric) 0.679883632....
Approximation (binomial) 0.6795
b) What is the exact probability that at most one of them was educated in Eurelia Use the approximate distribution that to get an approximation to the exact answer as well.
Exact (hypergeometric) 0.962191038
Approximation (binomial) 0.96185
6. a) The number of computer interruptions per day at a plant has (at least
approximately) a Poisson distribution. Last year, in the 366 days of
operation of the plant, there were only 10 days with no computer
interruptions. What was the average (mean) number of interruptions
per day ? How many interruptions were there in all during the year ?
b) A new configuration of the computer key board has 10 of the 26 letters
of the alphabet in different positions from the standard keyboard, and
the rest are in the usual places. If a perfectly accurate touch typist types the word computer on the new keyboard without looking, what is the probability that there will be at least two mistakes in the resulting printed word ? It may be assumed that the typist does not know the new configuration.
7. a) The number of arrivals of cement trucks at a construction site in an hour may be
treated as having a Poisson distribution with a mean of 4.6 per hour. What is the probability that more than 2 trucks will arrive in a randomly-selected half-hour time period?
b) Show that the distribution of the number of trucks arriving in an hour is positively skewed.
8. A printer prints 80 characters per line. The printer is old, and sometimes makes mistakes in printing characters. Each character has the same probability of being misprinted. A random sample of 100 lines produced by the printer only has 7 lines with no misprints. About how many of the 100 lines do you expect to have precisely 1 misprint amongst the 80 characters?
9. The traffic lights near the home of A.J.Jones change colour at random times, so that Jones considers the number of changes in any given time period to have a Poisson distribution, with a mean of 20 changes per hour.
a) What is the probability of more than the mean number of changes in a nine-minute period?
b) What is the median number of changes in a nine-minute period?
c) A.J. Jones tests his theory by counting the number of changes in a randomly-selected nine-minute period every day for many days. What is the probability that the first time that there are more than the mean number of changes is the fourth day that Jones checks?
c) What is the probability of having to wait more than 3 minutes for the light to change?
10. The number of flaws in 10-metre lengths of cable has approximately a
Poisson distribution with a mean of 0.24 flaws per metre.
a) What is the probability that twenty metres of the cable have at least 4 flaws?
b) A skeptical engineering group counted the number of flaws per 10-metre length for 400 lengths. They found that 32 of the 400 lengths had absolutely no flaws at all. What was the estimated figure that the engineering group obtained for the mean number of flaws per 10-metre length?
11. 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.
12. 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?
13. a) In an office employees have been categorized according to
seniority and age. Some of the results are cross-tabulated below.
Management White-collar
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?
14. 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?
15. 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?
16. 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?