STA 103 Homework 4, due Wednesday, Nov 12
Note: These problems are not in the book – they are custom problems made by Erin.
E.1 Suppose a machine breaks down occasionally as a result of a particular part that wears
out, and supposed these breakdowns occur randomly and independently. The average
number of breakdowns per 8-hour day is four, and the distribution of the breakdowns
does not change.
a. Find the probability that no breakdowns will occur during a given 8-hour day.
b. Find the probability that at most two breakdowns will occur during the first hour
of the day.
c. What is the minimum number of spare parts that management should have on hand
on a given 8-hour day if it wants to be at least 90% sure that the machine will not
be idle at any time during the day because of a lack of parts?
E.2 A boat broker in Florida receives an average of 26 orders per year for an exotic model
of cruiser. Assuming that the demand for this model remains the same throughout the
year, what is the probability that the broker will receive the following? (Assume there
are 52 weeks in a year.)
a. exactly 1 order for this model in a given week.
b. exactly 2 orders for this model over a given two-week period.
c. exactly 4 orders for this model over a given two-week period.
E.3 The scheduling manager for a certain hydro-power utility company knows that there
are an average of 12 emergency calls regarding power failures per month. Assume that
a month consists of 30 days.
a. Find the probability that the company will receive at least 2 emergency calls during
a specified month.
b. Suppose the utility company can handle a maximum of 3 emergency calls per day.
What is the probability that there will be more emergency calls than the company
can handle on a given day?
E.4 Suppose that a car dealer has 20 cars of the same make and model on a particular lot,
and that 7 of them have a defect in the airbag. Assume that you are equally likely to
pick any car to drive, and you choose to test drive 3 of them (without replacement).
a. Write down the distribution function for X if X is the number of cars with defective
airbags out of the 3 sampled.
b. What is the probability that you drive two cars with defective airbags?
c. What is the probability that you drive at least two cars with defective airbags?
d. What is the expected value and standard deviation of X?
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E.5 John is trying to select fiscally responsible stocks, and has narrowed his investment
choices down to 5 stocks. Assume that 3 of the stocks will fail, and that John will
randomly select 2 stocks to invest in.
a. What is the probability that he selects one of the stocks that will fail?
b. What is the probability that both stocks he selects will be failures?
c. What is the expected number of stocks that will fail out of the two that he picked?
E.6 Suppose that customers arrive at a checkout counter at an average rate of two customers
per minute and that their arrivals follow an exponential distribution.
a. What is the probability that a customer will arrive within one minute?
b. What is the probability that a customer will arrive between 1 and 3 minutes?
c. What is the probability that no customer will arrive by 3 minutes?
d. What is the average time we expect to wait for a customer to arrive? The standard
deviation?
E.7 On average, a particular computer part fails every 3 years. Assume that their time to
failure is exponentially distributed.
a. What is the probability that a computer part lasts more than 4 years?
b. What is the probability that a computer part fails by year 3?
c. If a part has not failed by year 2, what is the probability it fails by year 4?
E.8 Suppose that the average arrival time of a delivery truck is 15 minutes. If two trucks
cannot arrive at once, and the trucks are independent, find the following:
a. the number of trucks expected to arrive in an hour.
b. the probability that a truck does not arrive by one hour.
c. the probability that a truck arrives in a half hour.
d. the standard deviation of arrival time.
E.9 Suppose the amount of gasoline sold daily at a service station is uniformly distributed
with a minimum of 2000 gallons and a maximum of 5000 gallons.
a. What is the probability density function for X if X is the number of gallons of
gasoline sold daily?
b. Find the probability that daily sales will fall between 2500 and 3000 gallons.
c. What is the expected value of daily sales?
E.10 If the amount of spam your email receives is uniformly distributed with a minimum of
5 per day and a maximum of 20 per day, find the following:
a. The expected value and standard deviation of amount of spam per day.
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b. The probability that you get at least 10 spam messages in a day.
c. The probability that you get at most 12 spam messages in a day.
d. The standard deviation of number of spam messages received per day.
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