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Question 1.12 [2, 3]
Catherine has a Gmail account and categorises her emails according to work and non-work related
emails. The probability that an email is a work-related email is 65.32%. Suppose furthermore it is
given that the probability that a work-related email received is a spam email is 15.34% and that if it
is a non-work related email that it is spam is 5.6%.
Calculate the following probabilities
a) That an email received by Catherine is a spam email.
b) Given that the email is spam what is the probability that it is a non-work-related email.
Question 1.11[4]
There are two bags of chocolates. Bag one has 5 Barones and 2 KitKats and Bag 2 has 2 Barones
and the 7 KitKats. A chocolate is selected at random from Bag one and added to Bag two. A
chocolate is now drawn randomly from Bag two. Given that the chocolate selected is a KitKat what
is the probability that the original chocolate drawn from Bag one was a Barone? Show all working
out.
Question 1.10 [2, 2, 3]
Due to the increased demand of student transport during the examination period, a local business
has decided to loan a fleet of ten taxis that they use for their employees. Of these ten taxis used to
pick up students, five taxis must be sent to the Walmer Area in Gqeberha, three taxis must be sent
to the Kariega, and two taxis must be sent to the Gqeberha train station. Assume that taxis are
allocated to the destinations at random.
a) If one of the taxis is in need of repair, what is the probability that it was dispatched to Walmer?
b) If three of the taxis are in need of repair, what is the probability that every destination (Walmer,
Kariega and train station) gets one taxi in need of repair?
Question 1.9 [2, 2, 3]
Chantelle has decided to sell baked biscuits to assist in the payment of her university fees. After
baking for hours and packing packets to sell, she finds that she has 9 biscuits left over. Of these 9
biscuits, 4 are chocolate biscuits, 3 are raisin and 2 are peanut butter. She thinks to herself that she
is going to use these 9 biscuits to assist her with understanding probability. She treats each biscuit
as being slightly different, however order of her selection is not important.
Suppose Chantelle selects 3 biscuits at random from the 9, help her answer the following questions:
a) Calculate the probability that of the 3 biscuits randomly selected, 1 is chocolate, 1 is raisin
and 1 is peanut butter.
b) Calculate the probability that only chocolate biscuits are selected
c) Calculate the probability that at least
Question 1.7 [2, 2, 2, 2]
If A and B are independent events with P A( ) 0.52 = and P B( ) 0.21 = , find the following:
a) PA B ( ) ∪
b) PA B ( ) ∩
c) PA B ( ) ∪
d) Are A and B disjoint events? Motivate your answer!
Question 1.8 [3]
A certain washing machine factory has found that 15% of its washing machines manufactured in the
factory break down and are returned in the first year of operation. Suppose that 32 machines are
purchased by a laundromat from this washing machine factory, find the probability that at least one
washing machine breaks down in the first year of operation?
Question 1.2 [2, 2, 2]
Suppose that we have two events A and B such that P A( ) 0.8 = and P B( ) 0.7 = .
a) Is it possible that PA B ( ) 0.1 ∩ = ? Explain your answer.
b) What is the smallest possible value of PA B ( ) ∩ ?
c) What is the largest possible value of PA B ( ) ∩ ?
Question 1.1 [2, 2, 2, 2, 2]
A local farmer plants a given number carrots on a certain number of days. We are looking at the
number of carrots the farmer can plant over two days. Suppose that the famers must plant at least
4 carrots on the first day, no more than 9 carrots on the second day and farmer has to plant more
carrots on the second day than the first day.
a) Determine the sample space of the experiment.
b) If each of the outcomes in (a) have equal probability of occurring find the probability of the
following events:
i. Event that there were 13 carrots in total planted over the two days.
ii. Event that an odd number of carrots were planted on the second day.
c) Are the events (i) and (ii) mutually exclusive? Motivate your answer!
d) Are the events (i) and (ii) statistically independent? Motivate your answer!
3. A product is manufactured by four factories A, B, C and D. The unit production costs in them are ETB 2, ETB 3, ETB 1 and ETB 5 respectively. Their production capacities are 50, 70, 30 and 50 units respectively. These factories supply the product to four stories, demands of which are 25, 35, 105, and 20 units respectively. Unit transportation cost in ETB for each factory to each store is given in the table below.
Stores
1 2 3 4
2 4 6 11
10 8 7 5
13 3 9 12
4 6 8 3
A
Factories B
C
D
Determine the transportation plan to minimize the total production-cum-transportation cost by using:
a. Vogel’s Approximation Method (VAM) for initial basic feasible solution
b. Find the optimal solution using Modified Distribution Method (MODI) method
2. A firm produces three products A, B, and C, each of which passes through three departments: Fabrication, Finishing and Packaging. Each unit of product A requires 3, 4 and 2 hours; a unit of B requires 5, 4 and 4 hours while each unit of product C requires 2, 4, 5 hours respectively in the three departments. Every day, 60 hours are available in fabrication department, 72 hours in the finishing department and 100 hours in the packaging department. If the unit contribution of product A is ETB 5, of product B is ETB 10, and of product C is ETB 8, determine the number of units of each of the products, which should be made each day to maximize the total contribution. Also determine if any capacity would remain unutilised.
a. Write the formulation for this linear program.
b. Solve the Linear programming problem using simplex method.
2. A firm produces three products A, B, and C, each of which passes through three departments: Fabrication, Finishing and Packaging. Each unit of product A requires 3, 4 and 2 hours; a unit of B requires 5, 4 and 4 hours while each unit of product C requires 2, 4, 5 hours respectively in the three departments. Every day, 60 hours are available in fabrication department, 72 hours in the finishing department and 100 hours in the packaging department. If the unit contribution of product A is ETB 5, of product B is ETB 10, and of product C is ETB 8, determine the number of units of each of the products, which should be made each day to maximize the total contribution. Also determine if any capacity would remain unutilised.
a. Write the formulation for this linear program.
b. Solve the Linear programming problem using simplex method.
#331389 on Nov 2022