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Applied maths/AL/Tutorial/p&s-06/p.1
PLK Vicwood K.T.Chong Sixth Form College
Applied Mathematics (AL)
Tutorial
Topic: Normal distribution
Code:P&S-6
1. The weekly sales of petrol at a garage have a normal distribution with mean 50 kilolitres
and standard deviation 20 kilolitres and are assumed to be independent of each other.
(a) Find the probability that the sales of petrol are less than 25 kilolitres in a given week.
(b) It is found that even if the storage tank is full at the beginning of a week, there is a
probability of 0.04 that all the petrol will be sold out before the end of the week.
Determine the capacity of the storage bank.
(c) After an increase of petrol duty, the sales of petrol at the same garage are, on the
average, less than 25 kilolitres in 3 out of 7 weeks and more than 85 kilolitres in 1 out
of 40 weeks. Determine the new mean and standard deviation, to the nearest kilolitre,
of the weekly sales assuming that the sales still have a normal distribution.
(d) Assuming the situation in (c). If there are 80 kilolitres of petrol in stock when an order
for a new supply is placed and if delivery takes two weeks, find the probability that
the garage will run out of petrol before the next delivery. You may assume that the biweekly sales of petrol still normally distributed.
2. An organization has c parking spaces for its employees. N of its employees own cars, and,
on a weekday, each of them independently has a probability p of driving to the office.
(a) If c = 10, N = 12 and p = 0.6, find the probability that all parking spaces will be full on
a weekday.
(b) If c = 10, N = 20 and p = 0.6, use the normal approximation to find the probability that
all parking spaces will be full on a weekday.
(c) If N = 20 and p = 0.6, use the normal approximation to determine the minimum
number of parking spaces in order that on any weekday the probability that at least one
employee cannot find a parking space for his car is less than 0.10.
(d) If N = 20 and c = 12, use the normal approximation to determine the maximum value
of p if the probability that at least one employee cannot find a parking space for his car
on a weekday is less than 0.14.
Applied maths/AL/Tutorial/p&s-06/p.2
3. A suitcase storage system has a number of square compartment openings each with side
0.75 m. Records show that the widths and heights of suitcases are normally distributed.
The two distributions are independent of one another. The widths have a mean of 0.50 m
and a standard deviation of 0.245 m and the heights have a mean 0.60 m and a standard
deviation of 0.24 m. (Assume that the length of a suitcase is always at least 1 m.) A
suitcase is regarded as oversize if its width or its height is more than 0.75 m.
(a) Calculate the probability that a suitcase is oversize.
(b) Suppose that a suitcase will not be accepted for storage if it is oversize or overweight.
Suppose further that 20% of the suitcases are overweight and then 50% of the oversize
suitcases are also overweight.
(i) What is the probability that a suitcase will be accepted?
(ii) Calculate the probability that out of 100 suitcases, 70 or more will be accepted,
approximating by normal distribution if necessary.
4. A man has five coins, four of which are fair. The fifth one is suspected of having a
probability p (p  0.5) of falling heads. The following trial is carried out 100 times. At
each trial, a coin is chosen at random from the five coins and then tossed. After recording
the result, the coin is replaced and the trial is repeated. If he gets less than 56 heads, he
will conclude that all five coins are fair. Answer the following questions using normal
approximation to binomial distribution.
(a) Suppose that the fifth coin is in fact fair. What is the probability that the man will
conclude from his experiment that the fifth coin is biased?
(b) Suppose that in fact p = 0.8. What is the probability that the man will conclude that all
five coins are fair?