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Probability

Unit: Quantitative Analysis

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August 2025

 5 Questions
Question 1b
​​
Customers arrive at a particular supermarket exit at the rate of 4 customers per minute. The distribution of arrivals approximate a Poisson distribution. 
 
Required: 
Determine the probability that: 
 
(i) No customer arrives in a particular minute. 
 
(ii) At least one customer arrives during a particular minute. 
 
(iii) At most one customer arrives during a particular minute.


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Question 2c
​ ​ ​​In a certain college, 30% of the students failed in statistics and 15% failed in mathematics while 10% failed in both statistics and mathematics in a given semester.

Required: 

If a student is selected randomly, determine the probability that: 

(i) The student failed in statistics given that he failed in mathematics. 

(ii) The student failed in statistics or mathematics. 

(iii) The student failed in neither statistics nor mathematics. 


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Question 3b
​ ​​The probability of making a type I error on a certain washing machine is 0.10. The probability that the machine will remove all stains on a garment when there is a type I error is 0.30 and the probability that the machine will remove all stains on the garment when there is no type I error is 0.80. 
 
Required: 
Find the probability of type I error assuming that the machine fails to remove the stains. 


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Question 7b
​ ​ ​​The wages paid to workers of Baraka Ltd. are assumed to be normally distributed with a mean of Sh.37,000 and a standard deviation of 1,400. 
 
Required: 
Calculate the probability that an employee chosen at random earns salary of: 
 
(i) More than Sh.39,000.
 
(ii) Less than Sh.33,500. 
 
(iii) Between Sh.34,000 and Sh.40,000. 


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Question 4a
​ ​ ​ ​​The use of internet has become a major component of a student’s day. A research study classified use of internet into 3 categories for students: No use, light use and heavy use. The study observed 1,000 students over a period of one month and developed the following transition matrix of probabilities from day to day:


No use
Light use
Heavy use

No use
0.35
0.15
0.50
Previous day
Light use
0.30
0.35
0.35
Heavy use 
0.15
0.30
0.55

Required: 

 Suppose that the initial distribution for the three categories is (0.20, 0.40, 0.40), find the distribution after: 

 (i) 1 day. 

(ii) 2 days. 

(iii) At steady state. 


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April 2025

 2 Questions
Question 6a
​​Explain the meaning of the following terms as used in set theory: 
 
(i) Universal set. 
 
(ii) Subset. 
 
(iii) Set union. 


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Question 2a
​​Explain the following terms as used in probability theory: 
 
(i) Conditional probability. 
 
(ii) Bayes theorem.


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December 2024

 5 Questions
Question 3b
​ ​ ​ ​ ​ ​​The manager of Glaze Ltd. has selected a random sample of 150 invoices from the company’s outstanding invoices and provides the following distribution:

Sales
Sh.“000” 
Probability of
outstanding invoice 
0  –    100
0.20
100 – 200
0.18
200 – 300
0.22
300 – 400
0.16
400 – 500 
0.09
500 – 600
0.08
600 – 700 
0.04
700 – 800 
0.03

Required: 

 (i) Calculate the expected mean of the random sample.

(ii) Calculate the expected standard deviation of the random sample. 

(iii) Calculate the coefficient of variation of the random sample. 

(iv) Determine the 95% confidence interval of the mean value of the outstanding invoices. 


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Question 7a
​​Tangles Limited is the sole producer of three hair products; kinky, fluffy and flat that currently have a market 
share of 30%, 50% and 20% respectively. 
 
Each month, some brand switching takes place as follows: 
 
  • Customers that bought kinky the previous month, 70% buy it again while equal proportion switch to fluffy and flat respectively. 
  • Customers that bought fluffy the previous month, 60% buy it again while 25% switch to kinky and 15% to flat. 
  • Customers that bought flat the previous month, 70% remain while 10% switch to kinky and 20% to fluffy. 
 
Required: 

(i) Construct a probability transition matrix of the switching probabilities. 
 
(ii) Calculate the new market share a month after the current market share.
 
(iii) Calculate the steady state.


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Question 2a
​ ​ ​​Outline FOUR characteristics of a poisson probability distribution.


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Question 2b
​​In relation to probability theory, distinguish between “subjective approach to probability” and “classical approach to probability”.


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Question 2c
​ ​​A manufacturing company is evaluating the preferences of 250 customers regarding three new products; product Exe, product Wye and product Zed.  The following information was gathered from a sample survey: 
 
  • 35  customers prefer both product Exe and product Wye. 
  • 120 customers prefer either product Exe or product Wye but not product Zed. 
  • 50 customers prefer product Wye but not product Zed or product Exe. 
  • 140 customers prefer either product Wye or product Zed but not product Exe. 
  • 70 customers prefer either product Zed but not product Exe or product Wye. 
  • 20 customers prefer both product Exe and product Zed but not product Wye. 
  • 45  customers prefer product Exe but not product Wye or product Zed.  
  
Required: 
(i) Visualise the customers’ preferences using a Venn diagram. 
 
(ii) Determine the probability that a customer selected at random expressed a preference for all the three products.
 
(iii) Determine the probability that a customer selected at random expressed a preference for at most two of the three products.
 
(iv) Determine the probability that a customer selected at random did not prefer any of the three products.


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August 2024

 5 Questions
Question 7b
​​Highlight SIX assumptions of Markov analysis.


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Question 6a
​ ​​A random variable is normally distributed with a mean of 25 and a standard deviation of 5.

Required:

Using normal distribution approach:

(i) Calculate the value that will be exceeded 10% of the time assuming an observation is randomly selected from the distribution.

(ii) Compute the value that will be exceeded 85% of the time assuming an observation is randomly selected from the distribution.

(iii) Determine the two values of which the smaller value has 25% of the values below it and the larger value has 25% of the values above it.

(iv) Determine the value in which 15% of the observations will be below the distribution.


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Question 4a
​ ​​The disease monitoring centre has established that the annual number of pandemic occurrence follows a Poisson distribution with a mean of 0.30 per annum. 

Required: 

(i) The probability that no pandemic will occur within a year. 

(ii) The probability that at most two pandemics will occur within a year.


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Question 5a
​ ​​Describe TWO laws of probability.


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Question 2a
​ ​ ​​In a classroom with 15 students, 4 of whom are girls, the teacher randomly selects 4 students for a group project.

Required:

Determine the probability that the group of the students selected will be composed of the following:

(i) 3 girls and 1 boy. 

(ii) All boys. 

(iii) At least one girl.


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April 2024

 6 Questions
Question 7b
​ ​ ​​Agritech Consultants have engaged 3 farming trainees; Jim, Ken and Lorn. After six months, the Consultancy
supervisor noted that 5%, 7% and 9% of land planted by Jim, Ken and Lorn respectively had ungerminated plants.
The supervisor had distributed 40%, 25% and 35% of the land respectively.

Required:

(i) Present the above information in a tree diagram.

(ii) The probability of land with ungerminated plants. 

(iii) The probability that land with ungerminated plants was planted by either Ken or Lorn


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Question 7a
​ ​ ​​A training institution requires that all its employees participate in one of three sports clubs namely; football, rugby or chess. In each term, the training institution designates an open selection period within which employees may change from one club to another. 

Prior to the last open selection period, 10% of the employees preferred football, 35% preferred rugby and the remaining chess. During the open selection period, 25% of the employees taking football switched to rugby, while 15% switched to chess. 30% of the employees taking rugby switched to football and 10% switched to chess, 20% of the employees taking chess switched to football and 10% switched to rugby. 

Required: 

(i) The transition matrix. 

(ii) The percentage of employees that will be taking each sport after the last open selection period. 

(iii) Assuming that the trend continues, determine the percentage of the employees who will be taking each sport in the long-run.


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Question 6a
​ ​​A manager must select 4 employees for a job promotion. 12 employees are eligible for job promotion.

Required:

(i) Determine the number of ways in which 4 employees could be chosen.

(ii) Determine the number of ways in which the 4 employees could be chosen from the 12 employees in the department.


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Question 4b
​ ​ ​​
A research company has established that the occurrence of a contagious disease follows a Poisson distribution with a mean of 0.3 per week. 

Required: 

(i) The probability that no case of a contagious disease is reported. 

(ii) The probability that almost one case of a contagious disease is reported.


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Question 1c
​​Given the following sets: 

A = {a, b, c, d} 
B = {c, d, e, f, g} 
C = {h, i, j, c, d} 

Required: 

Find: 

(i) The universal set “⋃”. 

(ii) A⋂B⋂C. 

(iii) C' .


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Question 1a
​​State FIVE characteristics of binomial distribution


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December 2023

 1 Questions
Question 5b
​ ​ ​​Viwanda Ltd. produces light bulbs that are packed into boxes of 100. The company’s quality control department indicates that 0.5% of the light bulbs produced are defective.

Required:

(i) The percentage of boxes that will contain no defective light bulbs.

(ii) The percentage of boxes that will contain two or more defective light bulbs


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August 2023

 3 Questions
Question 4a
​​Explain the following terms as used in Markovian analysis:

(i) Transition matrix.

(ii) Equilibrium state.

(iii) Initial probability vector.


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Question 5a
​ ​ ​​The output of an acre of land is assumed to be normally distributed with an average of 52 bags of maize and a standard deviation of 3.2 bags.

Required:

The probability that the output of an acre of land:

(i) Is greater than 48 bags.

(ii) Is greater than 60 bags. 

(iii) Is less than 45 bags.

(iv) Lies between 50 bags and 60 bags. 


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Question 7c
​ ​ ​​A mobile phone manufacturer orders for a special component called PH-2 from four different suppliers; ​\(S_1, S_2, S_3\)​ and ​\(S_4\)​. 20% of the components are purchased from ​\(S_1\)​, 10% from ​\(S_2\)​, 30% from ​\(S_3\)​ and the remainder from ​\(S_4\)​.

From past experience, the manufacturer knows that 2% of the components from ​\(S_1\)​ are defective, 4% of the
components from ​\(S_2\)​ are defective, 3% of the components from ​\(S_3\)​ are defective and 1% of the components from ​\(S_4\)​ are defective. All components are placed directly in the store before inspection. A worker selects a component for use and finds it defective.

Required:
(i) The probability that the component was supplied by ​\(S_1\)​.

(ii) The probability that the component was supplied by ​\(S_2\)​ or ​\(S_4\)​


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April 2023

 3 Questions
Question 7a
​​Explain the meaning of the following terms in the context of probability theory:

(i) Mutually exclusive events.

(ii) Independent events.


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Question 1a
​ ​​A survey was conducted to establish the number of faulty components from a production process. The results indicated that 0.01% of the components produced were faulty. Each machine produces 10,000 components.

Required:

The probability of there being 3 or more faulty components assuming a Poisson probability distribution.


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Question 2a
​ ​ ​​A survey was recently conducted to determine the preferences of 360 customers with regard to three brands of cooking oil namely; sunflower oil, coconut oil and olive oil.

The following results were obtained:

  • 220 customers preferred sunflower oil.
  • 160 customers preferred coconut oil.
  • 180 customers preferred olive oil.
  • 80 customers preferred both sunflower oil and coconut oil.
  • 110 customers preferred both sunflower oil and olive oil.
  • 100 customers preferred both coconut oil and olive oil.
  • 50 customers preferred none of the brands of cooking oil

Required: 

(i) Present the above information in the form of a Venn diagram. 

(ii) Determine the probability that a customer picked at random prefers all the three brands of cooking oil. 

(iii) Determine the probability that a customer picked at random prefers at least two brands of cooking oil.


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December 2022

 3 Questions
Question 2b
​ ​​An examination was administered to a group of students and the results were as summarised below:

Result
% of candidates 
Passed with distinction
10%
Passed with credit
60%
Failed
30%

A candidate fails the examination if he/she obtains less than 40% in the examination. In order to pass with distinction, the candidate must obtain at least 75% in the examination. 

Required: 

Calculate the mean and standard deviation of the distribution of marks assuming that the marks scored are normally distributed.


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Question 3a
​​Distinguish between the following terms as used in probability:

(i) “Conditional probability” and “marginal probability”.

(ii) “Discrete probability distributions” and “continuous probability distributions”.


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Question 5b
​​State FOUR characteristics of the binominal distribution


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August 2022

 3 Questions
Question 7a
​ ​ ​​A technician at Light Industries Ltd. has established that the probability of a production process producing defective output is 0.2. A total of 60 units are produced from the process in a certain production period.

Required:

(i) The probability that exactly 10 of the units will be defective assuming a poisson distribution.

(ii) The probability that exactly 10 of the units will be defective assuming a binomial distribution.

(iii) The expected number and standard deviation of units expected to be defective assuming a binomial
distribution.


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Question 4a
​ ​ ​ ​ ​​A proposal to teach longer hours on weekdays rather than have weekend classes was put forward by a subject lecturer to his students. 

The following results were obtained:

Opinion
Students gender
In favour
Opposed
Undecided
Male
40
10
15
Female
20
30
20

Required: 

Calculate the probability that a student selected at random will be: 

(i) Female and in favour of the proposal. 

(ii) Either male or opposed to the proposal. 

(iii) Undecided given that the student is female.


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Question 6c
​ ​ ​ ​ ​​The data below shows the probability distribution of profits earned by firms in the manufacturing industry:

Profit Sh.“million”
Probability
10 – 20
0.05
20 – 30
0.05
30 – 40
0.10
40 – 50
0.15
50 – 60
0.30
60 – 70 
0.10
70 – 80
0.20
80 – 90
0.05

Required:

(i) The expected profit. 

(ii) The expected standard deviation. 

(iii) The coefficient of variation.


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April 2022

 4 Questions
Question 1b
​ ​​Explain the following terms as used in probability:

(i) Joint probability.

(ii) Mutually exclusive events.

(iii) Conditional probability.

(iv) Dependent events.


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Question 6b
​​State any four assumptions of the Poisson probability distribution.


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Question 4b
​ ​​The savings accounts in a certain microfinance bank have an average balance of Sh.240,000 and a standard deviation of Sh.60,000. The account balances are assumed to be normally distributed.

Required:

(i) The proportion of savings accounts whose balances are above Sh.275,000. 

(ii) The proportion of savings accounts whose balances lies between Sh.190,000 and Sh.260,000.


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Question 1c
​​A firm has 500 employees out of whom, 2% have a minor accident in a given year. Out of the employees who have a minor accident in a given year, 30% had safety instructions. 80% of all employees had no safety instructions. 

Required: 

The probability of an employee being accident free given that the employee had no safety instructions.


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Pilot December 2021

 1 Questions
Question 2
​ ​ ​​An electronics dealer in Nakuru has labelled a certain electrical component with numbers 1 – 50. These components are normally sold to 5 specific customers who pick one each on week days only. Incidentally, the components labelled numbers 3, 18, 12, 26 and 46 are defective.

Required:

(a) The probability that one customer will have drawn five defective components by the end of 5 weeks.

(b) The probability that two customers will have drawn two defective components each, two none and the other components, in two weeks.


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December 2021

 2 Questions
Question 6b
​​A global conference on "the blue economy" was recently held in Kenya and was attended by 280 delegates from America, Europe and Africa.

The following information relates to the delegates who attended the conference:

  • 70 delegates represented Europe
  • 96 delegates represented Africa
  • 128 delegates represented America
  • 20 delegates represented all the three continents.
  • 25 delegates represented America and Africa
  • 22 delegates represented America and Europe
  • 26 delegates represented Europe and Africa
Required:

(i) Present the above information in the form of a Venn diagram.

(ii) The number of delegates who represented at least two continents.

(iii) The number of delegates who represented only one continent.

(iv) The number of delegates who represented none of the three continents.


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Question 6c
​ ​​During the manufacture of a product, 0.002 of the product turns out to be defective. The product is supplied in packets of 10. A consignment of 100,000 packets is produced in a certain period.

Required:

Using the Poisson distribution, calculate the approximate number of packets containing:

(i) No defectives.

(ii) 1 defective.

(iii) 2 defectives.


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September 2021

 1 Questions
Question 5b
​​A recent inspection of bolts produced by a certain company revealed that 16 bolts were defective out of a total of 40 bolts inspected. 

5 bolts are picked at random and inspected. 

Required: 

(i) Assuming that the distribution of defective bolts follows the poisson distribution, calculate the probability that at least three bolts are defective. 

(ii) Assuming that the distribution of defective bolts follows the binomial distribution, calculate the probability that at most 3 bolts are defective.


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May 2021

 2 Questions
Question 2a
​ ​​Highlight properties of each of the following probability distributions:

(i) Binomial distribution.

(ii) Poisson distribution. 


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Question 1c
​ ​ ​​The Registrar of Highfliers University has observed that the grade point aggregate of the University's students is normally distributed with a mean of 2.75 and a standard deviation of 0.40. 

Required:

(i) The probability that a randomly selected student from the university has a grade point aggregate of between 2.00 and 3.00.  

(ii) The lowest grade point aggregate that should be obtained by a student for him/her to be among the top ten per cent of the students.  

(iii) Assuming that the university has a total of 10,000 students, determine the number of students having a grade point aggregate of 3.70 or higher.


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November 2020

 1 Questions
Question 2c
​ ​​In a certain hospital, the arrival rate of patients into the outpatient department is 3 patients per hour and 4 patients are normally attended per hour.

Required:
(i).    Service rate.
(ii).   Length of queue.
(iii).  Length of the system.
(iv).  The time a patient takes being actually attended.
(v).   The probability that there are more than six patients in the outpatient hospital department.


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November 2019

 1 Questions
Question 1b
​​Six consultants work for XYZ Ltd. A consultant has a 20% chanice of being absent from work in a given day. The company needs to establish the probability of more than two consultants being absent from work. 

Required: 
Compute the above probability of absence assuming: 
 (i) A binomial distribution.
(ii) A Poisson distribution.


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May 2019

 3 Questions
Question 1c
​​A medium sized commercial bank has a clientele of 200 active customers. The bank operates three different types of accounts namely; current account, savings account and fixed deposit account. Information obtained from the bank indicates that: 

  • 84 customers operate savings accoynts. 
  • 109 customers operate current accounts. 
  • 106 customers operate fixed deposit accounts. 
  • 45 customers operate both savings and current accounts. 
  • 36 customers operate both savings and fixed deposit accounts. 
  • 43 customers operate both fixed deposit and current accounts. 

 Required: 
(i) Present the above information in a venn diagram. 
(ii) The probability that a customer selected at random operates all the three types of accounts. 
(iii) The probability that a customer selected at random operates only two types of accounts.


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Question 2a
​​Enumerate four assumptions of:

 (i) A normal distribution. 
(ii) A binomial distribution.


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Question 2b
​​A certain store has three cashiers serving customers at any given point in time. Each of the cashiers can serve on average 5 customers per hour. The arrival rate of customers averages 12 customers per hour. 

Required: 
The probability that there are no customers in the queuing system at a given point in time.


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November 2018

 2 Questions
Question 1d
​​Bidii College offers three courses namely; Accounting, Computing and Driving. The college has a total population of 500 students. Data obtained from the college revealed the following: 

329 Students were undertaking Accounting course. 
186 Students were undertaking Computing course. 
295 Students were undertaking Driving course. 
83 Students were undertaking both Accounting and Computing courses. 
217 Students were undertaking both Accounting and Driving courses. 
63 Students were undertaking both Computing and Driving courses. 

Required: 
(i) Present the above information in a Venn diagram. 
(ii) The number of students undertaking all the three courses. 
(iii) The number of students undertaking only one course.


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Question 5a
​​Explain the following terms as used in probability theory: 

 (i) Mutually exclusive events. 
(ii) Independent events. 
(iii) Joint probability. 
(iv) Conditional probability.


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May 2018

 1 Questions
Question 5a
​ ​ ​​An electricity company has established that the weekly number of occurrences of lightning striking transformers follows a Poisson distribution with a mean of 0.4 per week.

Required:
 (i) The probability that no transformer will be struck by lightning in a week. 
(ii) The probability that at most two transformers will be struck by lightning in a week.


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December 2017

 3 Questions
Question 2c
​​A survey conducted on citizens of a certain country to determine the annual per capita income indicated that the annual income of the citizens is normally distributed with a mean of Sh.980,000 and a standard deviation of Sh. 160,000. One citizen was randomly selected from the country.

Required:
The probability that the annual income of the citizen: 
(i)   Is greater than Sh.500,000. 
(ii)  Is greater than Sh.1,220,000. 
(iii) Lies between Sh.852,000 and Sh.1,100,000.


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Question 1c
​​A survey was conducted on 800 houscholds to determine their preference for three consumer goods, namely Fex, Gex and Mex. The results of the survev were as follows: 

230 households preterred Fex. 
245 households preferred Gex. 
325 households preferred Mex. 
30 households preferred all the three goods. 
70 households preferred Fex and Mex. 
110 households preferred Fex only 
185 households preferred Mex only. 

Required:
 (i)  Present the above information in a venn diagram. 
(ii)  The number of households that preferred Fex and Gex. 
(iii) The probability that a household selected at random does not prefer any of the three goods.


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Question 2a
​​Enumerate three circumstances under which the Poisson distribution is most applicable.


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May 2017

 2 Questions
Question 2a
​​Highlight four properties of a binomial experiment.


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Question 2b
​​Viwanda Limited is a company that operates in the printing industry. The company has a total of 30 machines that operate a 24 hour cycle. The probability of a machine breaking down on any given day is 0.015. 

Required:
 (i).   The probability that exactly four machines break down in a given day. using poisson distribution. 
 (ii).  The probability that exactly four machines break down in a given day, using binomial distribution.
 (iii). Comment on the results obtained in (b)(i) and (b)(ii) above.


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November 2016

 2 Questions
Question 1b
​​The following information relates to product "X" which is susceptible to three types of defects; A, B and C. The probability of product "X" containing defect C depends on whether the product contains any other defects, A or B. The probabilities of the product containing the defects are as follows:
Type of defect
Probability
A
0.15
B
0.14
C(if it neither contains defect A nor defect B)
0.3
C(if it contains either defect A or defect B)
0.2
C(if it contains both defects A and B)
0.1

Required:
(i)      The probability that product "X" contains no defect.
(ii)     The probability that product "X" contains only one of the three defects


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Question 2b
​ ​ ​​Faidika College offers three courses, namely; Accounting, Information Technology and Statistics. The marketing department of the college conducted a survey on 500 students to determine the number of students enrolled for each of the three courses. The results of the survey were as follows:

  • 329 students were enrolled for Accounting.
  • 186 students were enrolled for Information Technology.
  • 295 students were enrolled for Statistics.
  • 83 students were enrolled for Accounting and Information Technology.
  • 217 students were enrolled for Accounting and Statistics.
  • 63 students were enrolled for Statistics and Information Technology.

Required:
(i) Illustrate the above information in a venn diagram.
(ii) The probability that a student is enrolled for all the three courses.
(iii) The probability that a student is enrolled for Accounting or Statistics but is not enrolled for Information Technology


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May 2016

 1 Questions
Question 2b
​​The mean weight of 500 packaging tins from a production process are normally distributed with a mean weight of 151 grammes and a standard deviation of 15 grammes.

Required:
(i) The number of packaging tins that weigh between 120 grammes and 155 grammes.
(ii) The number of packaging tins that weigh more than 185 grammes.


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November 2015

 1 Questions
Question 5b
The table below shows the probability distribution of the number of digital boxes sold by an electronics store on a daily basis:

Digital boxes sold (units)
Probability 		0
0.05 		1
0.05 		2
0.10 		3
0.15 		4
0.20 		5
0.15 		6
0.15 		7
0.10 		8
0.05

Required:
(i) The probability that the number of digital boxes sold in a given day is at least 3 but less than 7.

(ii) The mean daily sales of digital boxes.

(iii) The standard deviation of digital boxes daily sales.


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Pilot September 2015

 1 Questions
Question 5a
​​A machine is composed of three components X, Y and Z. The probability that component X is in good working condition is 7/10. If component X is in good working condition, the probability that component Y is in good working condition is 3/5. If component X is not in good working condition, the probability that component Y is in good working condition is 1/3. If components X and Y are in good working condition, the probability that component C is in good working condition is 5/6 otherwise, it is 1/10.

The machine can only be effective when component Z is in good working condition.

Required:
(i) The probability that the machine is effective.
(ii) The probability that only one component Y or Z is in good working condition.
(iii) The probability that component Y is in good working condition given that component Z is in good working condition.


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