F7 Mathematics and Statistics Chapter 6 Some Special Discrete Distributions
F7-MS-Ch6-2
Supplementary exercise
Geometric Distribution
Example
Example
20 % of the fish in a lake are known to be infested by a particular type of parasite.
A bag contains 8 gold coins and 2 silver coins, all are identical except in colour. One coin is
Let X be denoted the number of catches until the first infested fish is caught. Find
drawn at a time and if it is a gold it is replaced in the bag, if it is a silver coin it is set aside.
(a) the probability distribution for X
What is the average number of the drawings required to draw both the silver coins?
(b) E(X)
(c) var(X)
Example
Single item from the output of a machine are extracted and tested. It is known that 1 in 20 is
Example
faulty. Find
An urn contains 10 black balls and 6 white balls. Balls are random, one by one with
(a) the probability that the first faulty item found is the 6 th selected.
replacement, until a white ball has been obtained. Let X denoted the number of balls drawn,
th
(b) the probability that the first faulty item found comes before the 6 .
find E(X) and var(X).
(c) the expected number of items extracted before a faulty one is found.
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F7 Mathematics and Statistics Chapter 6 Some Special Discrete Distributions
F7-MS-Ch6-2
Supplementary exercise
Poisson Distribution
Example
A random variate has
Example
 =3 . Find
When a tyres manufacturing test new tyres by driving them over difficult terrain they find
(a) P(X=1)
that flat tyres externally caused occur on the average of once every 2000 miles. It is found
(b) P(X>3)
that also the Poisson process yield a useful model. What is the probability that
(c) E(X) and var(X)
(a) In a given 500 miles test no more that one flat will occur
(b) no flat in a trip of 4000 miles?
Example
It is found that on the average a computer makes a mistake every 20 minutes. If the mistake
occur at random intervals, calculate the probability of no mistakes occurring in an operation
Example
lasting for
The number of misprint on a page of a particular book is a Poisson variate with a mean of one
(a) 5 minutes
misprint per page. Find how many pages must be inspected so that there is a probability at
(b) 1 hour
least 90% of finding one or more misprints.
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F7 Mathematics and Statistics Chapter 6 Some Special Discrete Distributions
F7-MS-Ch6-2
Supplementary exercise
Example
It is known that on an average a computer makes a mistake every 10 minutes. If the mistake
Example
at random intervals, calculate the probability of
Telephone calls coming in to a switchboard follow a Poisson Distribution with mean 3 per
(a) no mistakes
minutes. Find the probability that in a given minutes there will be five or more calls.
(b) at least 2 mistakes
occurring in an operation lasting for 1 hour
Example
Customer at a certain department store enter at an average of 40 per hour
(a) Find the probability that no one enters the store during a particular five minute interval
Example
(b) Find the length of the time interval for which the probability that no one enters during is
In a certain football league it was noted after a long period of observation that the number of
0.5 .
goals score per match has a Poisson distribution , the mean number of goal per match being 3.
Calculate the probability that
(a) more than 4 goals will be score in a particular match
(b) at least two out of three matches will be goal-less
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F7 Mathematics and Statistics Chapter 6 Some Special Discrete Distributions
F7-MS-Ch6-2
Supplementary exercise
Example
Example
Of 150 football matches played last Saturday there will be 12 in which there were no score.
A company’s switchboard handles both internal and external calls. Past experience show that
Assuming a Poisson Distribution. What do you think the mean number of goals per match
from 10 a.m. to 11 a.m., the number of internal and external calls are independent and have
was?
Poisson Distribution with mean 3 and 5 respectively.
(a) Find the probability that on a particular day, there will be (I) no calls at all
(ii) a total of 3 calls from 10a.m. to 11 a.m.
(b) Given that a total of 3 calls arrived 10 a.m. to 11 a.m., find the probability that exactly 2
of them were internal calls.
Example
Suppose that the number of printing mistakes on each page of a 200-page book is
independent of that on the other pages and it follows a Poisson Distribution with mean 0.2
(a) Find the probability that on a given page contains one or more misprint.
(b) Find also the probability that the first misprint occurs on 11th page.
(c) Only pages containing misprint are returned to the printer for correction, find the
probability that a returned page contains only one misprint.
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F7 Mathematics and Statistics Chapter 6 Some Special Discrete Distributions
F7-MS-Ch6-2
Supplementary exercise
Binomial Distribution and Poisson Distribution
e.g.
Example 2
A book with 200 pages has 100 misprints. A page is selected at random, if the misprint on
Record show that customer complaints were received by a Public Relation officer at an
each page follow a Poisson Distribution, find the probability that it contains 1 misprint.
average rate of 3 per working day. The officer wants to find out the probability that he
receives more than 5 complaints in a day. Which distributions are most suitable for him to use.
Calculate the probability he wants.
Example1
If the probability is 0.2 that a marriage will end in divorce within 20 years after its start, find
Example3
the probability that out of five couples just married, in the next 20 years,
A traffic officer checks cars in succession. He knows from experience that the probability of a
(a) none of them will be divorced.
car not having a warrant of fitness is 1/30. Find the probability that
(b) at least 2 will be divorced if the number of divorce follows a distribution.
(a) the fifth car he checked is the first that do not have a warrant of fitness,
(b) of the five cars checked, exactly two do not have a warrant of fitness,
(c) of 90 cars checked in a week less than 4 do not have a warrant of fitness.
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F7 Mathematics and Statistics Chapter 6 Some Special Discrete Distributions
F7-MS-Ch6-2
Supplementary exercise
Example4
A manufacturing company sends out invoices to its customers allowing 5% discount to those
Example6
who settle their accounts within 15 days. In the past, 40% of the customers took advantage of
In a large consignment of packets of sugar, 0.5% of the packets are underweight.
the discount terms. On a particular day the company sent out 10 invoices. Find the probability
If 500 packets are examine, find the probability that
that less than 2 are settled within 15 days. For days when 10 invoices are sent out, calculate
(a) there are exactly 3 packets which are underweight.
the mean and variance of the distribution of the number of invoices settled within 15 days.
(b) none of the packets are underweight.
Example 5
A manufacturing if transistors claim that only 2% of his delivered lot of 3000 are imperfect.
To investigate this claim, the purchaser decides to test 20 these transistors chosen at random
Example7
and then accept the claim if at most one has imperfections, otherwise reject it. Find
Bacteria are distributed independently of one another in a solution and it is known that the
(a) the probability that the purchaser will reject the manufacturer’s claim even through it is
number of bacteria per cm3 follows a Poisson distribution with mean 0.6.
correct, and
(b) the probability that the purchaser will accept the manufacturer’s claim even through the
percentage defective is change to 5.
(a) Find the probability that a sample of 5 cm3 of solution contains 3 or more bacteria.
(b) Five samples, each of 5 cm3 of solution are taken. Find the probability that less than 2
these samples contain 3 or more bacteria.
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F7 Mathematics and Statistics Chapter 6 Some Special Discrete Distributions
F7-MS-Ch6-2
Supplementary exercise
Example8
The road accidents in a certain area occur at an average rate of one per two days. If the
Example9
occurrence of road accidents follow a Poisson Distribution. What is the most likely number of
At the ‘Hot Drink’ counter in a cafeteria both tea and coffee is sold. The number of cups of
accidents per week ? How many days in a week are expected to be free of accidents?
coffee sold per minute may be assume to have a Poisson distribution with mean 2 and the
number of tea sold per minute may be assume to be an independent Poisson distribution with
mean 1.5.
(a) Calculate the probability that in a given one-minute period
(i)
exactly 1 cup of coffee and 1 cup of tea are sold
(ii)
exactly 3 cups of hot drinks are sold
(b) In a given one-minute period exactly 3 cups of hot drinks are sold. Find the probability
Example9
that
A manufacturer of television set finds from experience that his sets on average have one
(i) these are coffee
major breakdown every four years. It is known that the number of breakdown follows a
(ii)
1 cup is coffee and 2 cups is tea.
Poisson Distribution.
(a) If he guarantee his sets for one year against a major breakdown, what proportion of
customers will be expected to use the guarantee for repairs?
(b) What guarantee would he need to give if he only wished 1 in 20 sets to be returned for
major repairs under guarantee?
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F7 Mathematics and Statistics Chapter 6 Some Special Discrete Distributions
F7-MS-Ch6-2
Supplementary exercise
8