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MTH219Fundamentals of Statistics and Probability

Tutor-Marked Assignment

January 2023 Presentation

MTH219

Tutor-Marked Assignment

TUTOR-MARKED ASSIGNMENT (TMA)

This assignment is worth 16% of the final mark for MTH219 Fundamentals of Statistics and

Probability.

The cut-off date for this assignment is 1 March 2023 (Wednesday) , 23 55 hours.

Note to Students:

You are to include the following particulars in your submission: Course Code, Title of the

TMA, SUSS PI No., Your Name, and Submission Date.

For example, ABC123_TMA01_Sally001_TanMeiMeiSally (omit D/O, S/O). Use underscore

and not space.

Question 1

Events A and B are such that P( A) = 0.4 , P( B | A) = 0.7 and P ( AC B ) = 0.3 .

(a)

Find the values of P( A B) and P(B) .

(4 marks)

(b)

Demonstrate, with reasoning, whether A and B are:

(i)

mutually exclusive,

(ii)

independent.

(4 marks)

(c)

A data set presented in the form of a frequency table is given as below:

Frequency

0

1

2

3

4

5

6

7

8

9

10

11

12

Data

2

180

904

7639

19764

22636

23710

23594

44868

57901

11511

83000

393

Table Q1(c)

SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)

Page 2 of 6

MTH219

Tutor-Marked Assignment

Determine the following quantities for the above data set:

(i)

Sample size;

(ii)

Mode;

(iii)

Median;

(iv)

Mean;

(v)

Inter-quartile range;

(vi)

Standard deviation;

(vii)

Range;

If you used Excel/R to aid in the computations, do include screenshots of the software

as proof.

(12 marks)

Question 2

John goes to the casino and plays a slot machine. The probability that he wins on the first spin

5

2

is . For all subsequent spins, the probability of John winning will be

if John wins in the

6

5

1

preceding round, and the probability of John winning will be if John did not win in the

5

preceding round.

(a)

John plays 3 rounds. Find the probability that he wins on the third round, given that he

only won two rounds of the three.

(4 marks)

(b)

Let X denote the number of rounds John need to play before he finally wins for the

first time. Comment on the suitability of modelling X after the geometric distribution.

Compute P( X = 5) .

(3 marks)

(c)

John visits the casino on 20 separate days, he played exactly 10 rounds on each day.

Let Y denote the number of days (out of 20) that John does not win anything.

State any necessary assumptions required in order to suitably model Y after the

binomial distribution. State clearly the parameters of this binomial distribution as well.

(4 marks)

(d)

Assume that your assumptions in Question 2(c) hold. Compute E(Y ) and Var (Y ) .

(4 marks)

SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)

Page 3 of 6

MTH219

Tutor-Marked Assignment

Question 3

Rachel, Sam and Thomas applied for a common position at the company. Only one of them

will be hired for the position. The probabilities that each of them is hired for the role are given

1 2

4

by , and

respectively.

7 7

7

The probabilities that Rachel, Sam and Thomas can help boost the company’s revenue are

4 1

3

given by , and

respectively, if they are hired.

10

5 2

(a)

Find the probability that Thomas is selected for the position given that the company’s

revenue did not improve.

(5 marks)

(b)

Find the probability, p, that the company’s revenue will generally improve.

(4 marks)

Rachel, Sam and Thomas interviewed at another company for another position. The

probabilities that each of them is hired for the role are now given by x, 2x and 4x

respectively.

It can be assumed that the hiring process for each individual candidate are independent

of one another.

(c)

Suppose there is no limit to the number of hires the company can make (i.e. the

company can hire any one of them, any two of them, all three of them or no one at all).

Find an inequality that must satisfied by x.

(7 marks)

SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)

Page 4 of 6

MTH219

Tutor-Marked Assignment

Question 4

In a café, coffee and tea are on average sold at 3 cups and 1 cup respectively every 5 minutes.

Let C and T denote the number of cups of coffee and tea sold in an hour interval.

(a)

State any assumptions required for us to use the Poisson distribution to model the

random variables C and T .

Comment on how reasonable the assumption is in practice.

(4 marks)

(b)

Assume that we can model C and T after Poisson distributions. Find the probability

that a total of 45 cups of beverages (you may assume the only beverages sold in the

café are either coffee or tea) in a randomly chosen one hour interval.

State any assumptions you have made in your computations.

(4 marks)

(c)

It is known that from 15:00 to 16:00, 45 cups of beverages are sold. Find the probability

that the number of cups of coffee sold in this time period is between 30 to 33 cups,

inclusive.

State any assumptions you have made in your computations.

(6 marks)

(d)

Let W denotes the time (in hours) passed after a cup of beverage is sold before the next

is sold.

State the probability distribution of W and write down the values of E (W ) and Var (W )

(5 marks)

(e)

I am in a queue at the café waiting to order my drink. There are two counters, A and B,

in front of me that I can turn to. On average, I need to wait for 10 minutes for my turn

to arrive to order at counter A and 8 minutes for that at counter B.

I will proceed to whichever counter is available first. Find the probability that I will

have to wait for at least 7 minutes for my turn to make my order.

State any assumptions you have made in your computations.

(5 marks)

SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)

Page 5 of 6

MTH219

Tutor-Marked Assignment

Question 5

Safety checks on the roads outside the university are conducted on a daily basis. It is known

that on average, 0.5 potholes are found per day on the roads.

(a)

Find the probability that on a randomly chosen day, at least 2 potholes are found.

(4 marks)

(b)

Find the probability that in a randomly chosen period of 20 days, there are exactly 7

days in which at least 2 potholes are found.

(5 marks)

Cracks on the road are also found an average rate of 0.75 per day.

(c)

Find the probability that in a randomly chosen period of 20 days, there is at most 2

potholes found given that a total of 10 faults (cracks + potholes) are discovered. State

any assumptions you may have made.

(8 marks)

When a pothole or a crack is discovered, there is 0.4 chance that it will be repaired by the end

of the same day.

(d)

Find the probability on a randomly chosen day, the holes/cracks discovered will not be

able to all be repaired by the end of the day.

(8 marks)

Remark: For Question 5(d), you may use the following identity:

xk

x 2 x3

=

1

+

x

+

+ + … = e x .

2! 3!

k = 0 k!

—- END OF ASSIGNMENT —-

SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)

Page 6 of 6

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