Download Probability Distributions Assignment - Mr-Kuijpers-Math

Probability Distributions Assignment
1.
John has to pass through six sets of traffic lights on his way to work. If he has to stop at
more than four sets of light, then he will be late for work. The sets of traffic lights
operate independently of each other. The probability the John will have to stop at any
one of the sets of traffic lights is 0.4.
Find the probability that John will be late for work.
2.
A power company has found that 5% of its clients will leave the company in any given
year. It is assumed that one client leaving the company is independent of any other
client leaving the company. A sample of 15 clients is chosen at random from the
company’s database.
Four conditions must be satisfied for the number of clients leaving the company to be
modelled by a Binomial distribution. Two of the conditions are:
- a fixed number of trial (ie 15) and
- only two possible options, either a client leaves the company or does not.
Identify the other two conditions that must be satisfied.
a)
b)
Find the probability that there is exactly one client who will leave the power company in
the next year.
c)
Find the probability that there is more than one client who will leave the power company
in the next year.
3.
A clothing company has to make 10 of its 200 staff redundant. The company states that
it will randomly decide who is to be made redundant. 30% of the staff are over the age
of 50. Let the random variable R represent the number of workers over the age of 50
who are made redundant out of the 10 people made redundant. The distribution of R
can be approximated by a Binomial distribution.
The condition of constant probability of success necessary for R to be modelled by a
Binomial distribution is not fully me. Explain why this is so.
a)
b)
Find the probability that the first person chosen at random to be made redundant is over
the age of 50.
c)
If the process is truly random, of the 10 people selected to be made redundant, what is
the expected number that will be over the age of 50?
d)
When the redundancies are made, 8 out of 10 people made redundant are over the age
of 50. The 8 people decide to take the company to court arguing that the company did
not use a random method to select who is to be made redundant as it stated.
Assuming that this situation can be approximated by a Binomial distribution, find the
probability that out of the 10 people made redundant, 8 or more will be over 50.
e)
As the judge in this case, you are to decide if the actions of the company are truly
random or not. Write a brief statement justifying your decision.
4.
In an experiment involving 10 trials, a scientist found that the probability of three
successes was 0.2668. What was the probability of one trial being a success?
5.
On average, a busy office has four phone calls every five minutes. What is the
probability that the office will have three phone calls in a five-minute period?
6.
a)
The number of serious mountaineering accidents that occur in New Zealand follows a
distribution with an average of 30 serious accidents per year. Estimate the probability
that:
Fewer than 3 serious accidents occur in any two month period.
b)
There are exactly 3 serious accidents in one month.
7.
The number of vehicles that a local petrol station serves between 10 am and 2 pm can
be modelled by a Poisson distribution, with a mean of 12 vehicles per hour during this
time.
Find the probability that no vehicle is being served during any given 10-minute interval.
8.
Curtain material is produced in 10 m lengths. Faults occur independently in the material
at an average rate of 0.5 faults per 10 m length. A length of material is judged
unsatisfactory if it contains more than one fault.
Name the distribution which models the number of faults in a 10 m length of curtain
material, stating the value of the parameter of this distribution.
a)
b)
Find the probability that a 10 m length of curtain material has no faults.
c)
The curtain-making factory produces a large quantity of 10 m lengths in one batch.
Estimate the proportion of unsatisfactory 10 m lengths of curtain material in this large
batch.
9.
For a Poisson random variable X, the probability X is zero is 0.375. Find the standard
deviation of X.
10.
a)
Car rides use a mean of 7 litres of petrol with a standard deviation of 0.6 litres.
Assuming the amount of petrol used is Normally distributed:
What is the probability that the car uses more than 8.2 litres on a ride?
b)
What is the probability that a car ride uses between 6.4 and 7.6 litres?
c)
If the car used in the car rides usually has only 8 litres of petrol in its tank, how many
times in the next 1,000 rides will the car be expected to run out of petrol?
11.
Cargo containers are unloaded from a ship by a crane and stored on the wharf. The
weights of the containers are Normally distributed with means of 12 tonnes and
standard deviation of 3 tonnes. The maximum weight that the crane can lift is 16 tonnes.
If a container is heavier than 16 tonnes then a special heavy lift crane must be brought
over to the ship to lift the container. The ship has 250 containers on it.
How many of the containers will need to be lifted by the special heavy lift crane?
a)
b)
Moving the heavy lift crane costs a lot of money so the wharf company wishes to buy a
new crane that will allow them to move 98% of all containers without having to use the
special heavy lift crane. What should be the maximum capacity. M, of the new crane?
(Give your answer to the nearest tonne.)
12.
180 adult males were involved in a study which attempted to identify a relationship
between the brain size and their intelligence. 10% of these males had brains that were
estimated to weigh more than 1400 g. Assume that the brain weights of the 180 males
can be well modelled by a Normal distribution. Under this model, how many standard
deviations is a brain weight of 1400 g above the mean brain weight?
13.
John buys some pizza to feed his friends. The pizzas are considered to be underweight
if they weigh less than 300 grams. The weights of the pizzas are independent and
Normally distributed with a mean of 320 grams and a standard deviation of 20 grams.
What is the probability that the two pizzas bought for dinner are both underweight?
14.
As a present Elizabeth and John received a box of imported white chocolates. It is
known that the 12 chocolates in the box are independently selected. The individual
chocolates have weights that are Normally distributed with a mean of 64 grams and a
standard deviation of 3 grams. On the side of the box it states that the 12 chocolates in
the box have a total net weight of 750 grams.
What is the probability that the 12 chocolates have a total net weight of at least 750 g?
15.
The flight time from Sydney to Auckland on a Boeing 767 is approximately Normally
distributed with a mean of 183 minutes and a standard deviation of 7 minutes. The
corresponding flight time for an Auckland to Sydney flight is approximately Normally
distributed with a mean of 196 minutes and a standard deviation of 10 minutes.
Let the random variable T represent the total flight time of a Boeing 767 for three
Sydney to Auckland flights and two Auckland to Sydney flights.
Give an appropriate model for the probability distribution of the random variable T. Fully
justify your answer, stating clearly any necessary assumptions.
16.
The amount of milk in a container is Normally distributed, with mean 1010 mL and
standard deviation 5.5 mL. Find the probability that 5 such containers of milk contain
less than 5015 mL of milk. (Assume containers are independent.)
17.
a)
A packet of biscuits contains 20 biscuits as well as the packaging. The mass of
individual biscuits is Normally distributed with a mean of 12 grams and a standard
deviation of 0.5 grams. The mass of the packaging is Normally distributed with a mean
of 15 grams and a standard deviation of 2 grams. The masses of the biscuits in a
packet are assumed to be independent of one another.
Calculate the mean mass for a packet of biscuits.
b)
Calculate the standard deviation for the mass of a packet of biscuits.
c)
Find the probability that the mass of a packet of biscuits is more than 252 grams.
d)
What mass do 95% of packets of biscuits lie above?
18.
The Normal approximation to the Binomial distribution and the Normal approximation to
the Poisson distribution are both situations where a continuity correction should be
applied. What common feature of the Poisson and Binomial distributions makes this
appropriate?
19.
a)
An airline operates a Boeing 747 aircraft which has a maximum seating capacity of 324
passengers. On a certain sector on which this aircraft is used, on average, 4% of
booked passengers fail to turn up for the flight (‘no-shows’). In anticipation of these noshows, the airline has a policy of ‘overbooking’ by accepting bookings for up to 330
passengers on this sector.
Let X be the number of ‘no-shows’ for a flight on which 330 passenger bookings have
been made. A binomial distribution is used to model the distribution of the random
variable X.
State the parameters and their values for this Binomial model.
b)
Find the mean and standard deviation of the random variable X.
c)
Consider flights for which 330 seats have been booked. For what percentage of these
flights does everyone who turns up gain a seat? That is, find P(X ≥ 6). Use a Normal
approximation with a continuity correction.
d)
One of the assumptions underlying the Binomial model will almost certainly never be
satisfied in this situation. State this assumption and clearly explain clearly why it will
almost certainly never be satisfied here.