Topic: Probability

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  • 1 Marks

QTB – Nov 2014 – L1 – SA – Q3 – Probability

Determines the probability that daily sales exceed a specified value, given a normal distribution with known mean and standard deviation.

The daily sales figures of a supermarket follow a normal distribution with a mean of N60,000 and a standard deviation of N14,000. Find the probability that the sales figure of a certain day exceeds N46,000.
A. 0.1587
B. 0.1590
C. 0.8413
D. 0.8415
E. 0.8423

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  • 1 Marks

QTB – Nov 2014 – L1 – SA – Q2 – Probability

Determines the probability that a customer does not receive a mutilated note from the cashiers.

The following tree diagram shows the scenario with two paying cashiers (C1 and C2) at a Microfinance Bank where M represents mutilated notes and N represents new notes:

he probability that a customer of the bank does not receive a mutilated note is:
A. 0.1125
B. 0.8725
C. 0.8875
D. 0.8525
E. 0.8850

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  • 1 Marks

QTB – Nov 2014 – L1 – SA – Q1 – Probability

Determines the correct definition of mutually exclusive events.

Two events are said to be mutually exclusive if
A. The occurrence (or non-occurrence) of one event does not affect the occurrence (or non-occurrence) of the other event
B. Both events can occur simultaneously
C. The occurrence of one event precludes the occurrence of the other event
D. Both are impossible events

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  • 20 Marks

QT – May 2016 – L1 – Q4 – Probability

Determine percentages of receipts using normal distribution and calculate mean and standard deviation based on given conditions.

a) Receipts at a particular depot have amounts which follow the Normal distribution with a mean of GH¢103.60 and a standard deviation of GH¢8.75.

Required:
i) Determine the percentage of receipts over GH¢120.05.
ii) Determine the percentage of receipts below GH¢92.75.
iii) Determine the percentage of receipts between GH¢83.65 and GH¢117.60.
iv) Determine the receipts amount such that approximately 25 percent of receipts are greater.
v) Above what amount will 90 percent of receipts lie?

b) If 10.56 percent of receipts have an amount above GH¢110.05 and 4.01 percent of receipts have an amount above GH¢120.05.

Required:
Calculate the mean and standard deviation of the receipts assuming that they are normally distributed.

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  • 20 Marks

QT – May 2016 – L1 – Q3 – Probability

Calculate the probabilities of various outcomes in a card drawing scenario, including conditional probability based on drawing a red card.

a) If from a normal pack of 52 cards, consisting of four suites each of 13 cards, one card is randomly selected:

Required:

Calculate the probabilities of selecting the following:

i) An ace
ii) A club
iii) An ace or a club
iv) The ace of clubs
v) A picture card (i.e. a jack, queen or king)
vi) A red card
vii) A red king
viii) A red picture card

b) Given that a card selected is red, calculate the probability that it is a picture card.

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  • 20 Marks

QT – Nov 2017 – L1 – Q5 – Probability

Use the normal distribution to compute probabilities related to loan default amounts.

The distribution of the total loan amounts defaulted by customers of a bank annually is approximated by a normal distribution. The average default amount is GH¢1.50 million, and the standard deviation is GH¢0.50 million.

Required:
a) Determine the probability that the total loan amount defaulted exceeds GH¢1.50 million. (3 marks)
b) Determine the probability that the total loan amount defaulted is between GH¢0.86 million and GH¢0.90 million. (4 marks)
c) Determine the probability that the total amount defaulted is at most GH¢2 million. (4 marks)
d) Determine the amount to be allowed per annum for loan defaults if 1% of actual defaults exceed this amount. (3 marks)
e) Determine the lower quartile of the distribution of total loan amounts defaulted. (3 marks)
f) Determine the upper quartile of the distribution of total loan amounts defaulted. (3 marks)
(Total: 20 marks)

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  • 16 Marks

QT – May 2019 – L1 – Q6b – Probability

Calculate probabilities based on student survey data regarding their favorite sports.

A survey of The Institute of Chartered Accountants (Ghana) students asked the question: “What is your favourite sport?” The results are summarized below:

Level Football Boxing Hockey Total
1 68 41 46 155
2 84 56 70 210
3 59 74 47 180
Total 211 171 163 545

Required: i) What is the probability of selecting a student whose favourite sport is boxing? (2 marks)

ii) What is the probability of selecting a Level 1 student? (2 marks)

iii) If the student selected is a Level 2 student, what is the probability that the student prefers hockey? (3 marks)

iv) If the student selected is a Level 2 student, what is the probability that the student prefers football or hockey? (3 marks)

v) If the student selected prefers football, what is the probability that the student is a Level 1 student? (3 marks)

vi) If the student selected is a Level 3 student, what is the probability that the student prefers football, boxing, or hockey? (3 marks)

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  • 4 Marks

QT – May 2019 – L1 – Q6a – Probability

Define collectively exhaustive events and complement of an event in probability theory.

Define the following terms in probability theory:

i) Collectively exhaustive events (2 marks)
ii) The complement of an event (2 marks)

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  • 20 Marks

QT – May 2019 – L1 – Q4b – Probability

Solve problems involving normal distribution, life spans, and warranty calculation using Z-scores.

CarJoy Manufacturing Company produces electronic components that have life spans normally distributed with mean  hours and standard deviation σ hours. Only 1% of the components have a life span less than 3,500 hours and 2.5% have a life span greater than 5,500 hours.

Required:
i) By standardizing the values of  to the standard normal values, obtain TWO (2) linear equations in  and . (4 marks)

ii) Using an appropriate method of solving simultaneous equations, determine the value of  and  in (i) above. (4 marks)

iii) Determine the proportion of electronic components with life spans between 4,120.50 hours and 5,052.45 hours. (4 marks)

iv) If the company gives a warranty of 4,000 hours on the components, find the percentage of components the company is expected to replace under the warranty. (4 marks)

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  • 4 Marks

QT – May 2019 – L1 – Q4a – Probability

Sketch the normal distribution curve showing the mean, median, and mode.

The continuous random variable X is normally distributed with mean μ and variance σ2.

Required:
Sketch the distribution of  and indicate on your sketch the mean, median, and mode. (4 marks)

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  • 20 Marks

QT – May 2019 – L1 – Q3 – Probability

Draw a decision tree and determine the optimal strategy for managing the cost of an art exhibition.

The Business Manager of Omaya Art Gallery has rented a hall to display the artworks of the artists of the gallery. She is considering organizing an exhibition of a number of rare painting masterpieces. In the past, only 70% of the paintings were sold in the first week. Moreover, if no painting is sold in the first five (5) days, the exhibition could be extended for another two (2) days but only 20% of the paintings would be sold.

The cost of the exhibition is GH¢500 per day. The manager estimated that in case she does not make any sales, she will have to pay GH¢15,000 to cover the costs of renting the exhibition hall for the same period.

Required: a) Draw a decision tree representing the Business Manager’s decision-making process. (8 marks)

b) Calculate the expected monetary cost of each decision node. (6 marks)

c) Determine the Business Manager’s optimal strategy. (6 marks)

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  • 6.5 Marks

QT – May 2017 – L1 – Q6b – Probability

Determine the probabilities of specific outcomes when tossing three fair coins.

If three fair coins are tossed:

Required:

i) Determine the probability of getting two heads and a tail.
(3.5 marks)

ii) Determine the probability of getting three tails.
(3 marks)

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  • 13.5 Marks

QT – May 2017 – L1 – Q6a – Probability

Calculate probabilities from a tree diagram based on people's chocolate preferences and eye color.

Consider people’s preferences in terms of chocolate and their eye color, presented as a probability tree diagram below:

  • 30% of people have brown eyes, 40% have green eyes, and the remaining have blue eyes.
  • 70% prefer milk chocolate, 10% prefer white chocolate, and the remaining prefer plain chocolate.

Using the tree diagram above, calculate the probability that a person:

i) Prefers plain chocolate and has brown eyes.
(1.5 marks)

ii) Prefers milk chocolate and has brown eyes.
(1.5 marks)

iii) Prefers white chocolate and has brown eyes.
(1.5 marks)

iv) Prefers plain chocolate and has green eyes.
(1.5 marks)

v) Prefers milk chocolate and has green eyes.
(1.5 marks)

vi) Prefers white chocolate and has green eyes.
(1.5 marks)

vii) Prefers plain chocolate and has blue eyes.
(1.5 marks)

viii) Prefers milk chocolate and has blue eyes.
(1.5 marks)

ix) Prefers white chocolate and has blue eyes.
(1.5 marks)

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  • 5 Marks

QT – Nov 2016 – L1 – Q1c – Probability

This question deals with calculating the conditional probability of earning more than GHC 5,000 given that an ICAG-qualified individual stays at a university.

If selected by the panel, the probability that an ICAG-qualified member will remain with the Private University is 0.6, and the probability that a Chartered Accountant earns more than GHC 5,000 per month in the university is 0.5. If the probability that Mr. Agbagba will remain with the university or earn more than GHC 5,000 per month is 0.7:

Required:
Calculate the probability that he will earn more than GHC 5,000 per month given that he is a Chartered Accountant who will stay with the university.

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  • 9 Marks

QT – Nov 2016 – L1 – Q1b – Probability

This part focuses on calculating the probability of an ICAG-qualified individual and their spouse being selected for a position.

Mr. Agbagba, an ICAG qualified member, and his wife, an ICAEW qualified member, attended an interview for two vacancies for the post of College Finance Officer at a Private University. The probability of the interview panel selecting the man is 1/7, and that of the wife is 1/5.

Required:
Assuming the event of selecting a man and selecting a woman are independent, determine the probability that:
i) Both of them will be selected. (3 marks)
ii) Only one of them will be selected. (3 marks)
iii) None of them will be selected. (3 marks)

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  • 6 Marks

QT – Nov 2016 – L1 – Q1a – Probability

Explains key probability terms such as independent events, mutually exclusive events, and joint probability.

Distinguish between the following terms as used in probability:
i) Independent Events and Dependent Events. (2 marks)
ii) Mutually exclusive Events and Exhaustive Events. (2 marks)
iii) Marginal Probability and Joint Probability. (2 marks)

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  • 20 Marks

QT – Nov 2018 – L1 – Q7 – Probability

Calculate expected returns for investments, determine optimal strategy, and analyze student error distribution.

Royal Driving School is considering investing in a profitable project. The school is given the following investment alternatives and percentage rates of return.

Over the past 300 days, market conditions have been moderate for 150 days and good for 60 days.

Required:
i) Calculate the expected return for each type of investment. (4 marks)
ii) Determine the optimum investment strategy for Royal Driving School. (3 marks)

b) The number of errors made by 294 students of Royal Driving School in their first attempt at a driving test is grouped in the following frequency distribution:

Number of Errors Number of Students
7 – 13 3
14 – 20 12
21 – 27 23
28 – 34 44
35 – 41 54
42 – 48 56
49 – 55 43
56 – 62 24
63 – 69 23
70 – 76 12

Required:
i) Compute an estimate of the mean and mode for the distribution. (3 marks)
ii) Construct an ogive for the distribution. (4 marks)
iii) Using the ogive in (ii) above, estimate the median for the distribution. (3 marks)
iv) Use the ogive in (ii) above to estimate the percentage of errors within one standard deviation of the mean. (3 marks)

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  • 14 Marks

QT – Nov 2018 – L1 – Q5b – Probability

Solve normal distribution problems related to spending habits of residents in Kojokrom.

In a study commissioned by Ofo Stores, the researcher examined the spending habits of the residents of Kojokrom. He found the spending habits to be normally distributed with a mean of GH¢700 and a standard deviation of GH¢70.

Required:
i) Determine the probability that a resident selected at random spends:

  • Less than GH¢620 (3 marks)
  • More than GH¢1,000 (3 marks)
  • Between GH¢800 and GH¢900 (4 marks)

ii) Calculate the amount:

  • Above which 80% of the residents will spend in a week (2 marks)
  • Below which 30% of the residents will spend in a week (2 marks)

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  • 6 Marks

QT – Nov 2018 – L1 – Q5a – Probability

Define properties of normal distribution and examples of phenomena that follow it.

The normal distribution is a probability distribution which usually applies to continuous variables.

Required:

i) State FOUR (4) properties of the Normal Distribution. (4 marks)

ii) State TWO (2) examples of phenomena which closely follow a Normal Distribution. (2 marks)

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  • 5 Marks

QT – Nov 2015 – L1 – Q7a – Probability

Explain five key decision-making terms including Maximax rule and Expected Monetary Value.

Explain the following terms in decision making:
(i) Maximax Rule
(ii) Maximin Rule
(iii) Expected Monetary Value
(iv) Payoff Table
(v) Expected Value of Perfect Information

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