MTH219 SUSS Probability and Statistics: Applications in Real-World Scenarios, Singapore

Question 1

From a survey, it was found that about 40% of adults regularly consume beverage A, 60% regularly consume beverage B, and 75% regularly consume at least one of the two beverages.

(a) Calculate the probability that a randomly selected adult regularly consumes both beverages A and B?
(5 marks)

(b) Compute the probability that a randomly selected adult does not regularly consume at least one of the two beverages?
(5 marks)

(c) Compute the probability that a randomly selected adult does not regularly consume both beverages?
(5 marks)

(d) It was also found that half of the adults consume sandwich regularly, and 30% of the adults regularly consume both beverage A and sandwich. What is the probability that a randomly selected adult either regularly consumes beverage A or does not regularly consume sandwich?
(5 marks)

(e) Given that the probability that an adult regularly consumes beverage A, beverage B and sandwich is 0.1, and the probability of regularly consuming beverage B and sandwich but not beverage A is 0.15. Determine the probability that a randomly selected adult regularly consumes beverage B or sandwich?
(5 marks)

Question 2

(a) A tiling company produces tiles using three different machines M1,M2,M_1, M_2,M1​,M2​, and M3M_3M3​. As part of the company’s quality control efforts, tiles with defects will need to be modified before they are sold to customers. Based on historical records, 5% of the tiles produced by M1M_1M1​ need modification, 8% of M2M_2M2​’s tiles need modification, and 10% of M3M_3M3​’s tiles need modification. Of all the tiles produced by the company, 50% of them are from M1M_1M1​, 30% of them from M2M_2M2​, and 20% of them from M3M_3M3​.

(i) Compute the probability that a randomly selected tile needs to be modified.
(5 marks)

(ii) If a randomly selected tile needs modification, determine the probability that it is produced by machines M1,M2,M_1, M_2,M1​,M2​, and M3M_3M3​, respectively.
(6 marks)

(b) Of all the drivers who drove past a certain inspection point for drink-driving, 60% of them pass the alcohol breath test. Assume that successive drivers pass the test independently of each other.

(i) Compute the probability that all of the next 3 drivers examined pass the alcohol breath test.
(3 marks)

(ii) Compute the probability that at least one of the next three drivers examined fail the alcohol breath test.
(3 marks)

(iii) Compute the probability that exactly one of the next three drivers examined pass the alcohol breath test.
(3 marks)

(iv) Compute the probability that all of the next three drivers examined pass the alcohol breath test, given that at least one of them pass the examination.
(5 marks)

Question 3

(a) Two six-sided dice, each numbered from 1 to 6, are rolled and different prizes are given based on the combination outcome. The rules of the game are as follows:

Rule 1: If a pair, i.e. (1,1), (2,2) etc., is rolled, then the player wins an amount double of the sum of the rolls (i.e. if the player rolls 5-5, then he will win $(5+5)\times 2=20$ dollars).

Rule 2: If the combined roll is equal to or above 8, then the player wins an amount equal to the sum of the rolls, except when it is a pair (i.e. if the player rolls 3-6, then he will win $3+6=9$ dollars).

Rule 3: Otherwise, the player loses $Y$.
Note: In the game, the player wins $1 means the banker loses $1 and vice versa.

The banker would like to make an average of $1 profit per game. Determine the value of $Y$ so that this objective can be achieved. Round off to the nearest dollar.
(13 marks)

(b) The time interval $X$, in minutes, between fishes being spotted by a group of fishermen has the following cumulative distribution

F(x)={0,x≤0,1−e−6x,x>0.F(x) = \begin{cases} 0, & x \leq 0, \\ 1 – e^{-6x}, & x > 0. \end{cases}F(x)={0,1−e−6x,​x≤0,x>0.​

(i) What is the probability that the time interval between successive fishes being spotted to be at least 10 seconds?
(4 marks)

(ii) What is the probability that the time interval between successive fishes is between 1 minute and 2 minutes?
(4 marks)

(iii) Solve the probability density function of $X$.
(4 marks)

Question 4

A box contains many balls and the mass $X$ (in grams) of the balls are uniformly distributed over the discrete values 11g, 12g, 13g, 14g, 15g, 16g, and 17g.

(a) A ball is randomly picked from the box. Compute the probability that the selected ball has mass less than 12 g.
(3 marks)

(b) Determine the mean and variance of $X$.
(4 marks)

(c) In a game, a player must select 5 balls from the box. If the lightest ball has mass less than 12g, then the player wins.

(i) To find the probability that the player will win, it is suggested that a Geometric model can be applied to compute the probability that the first ball with mass less than 12g can be obtained in not more than 5 balls drawn from the box. Do you agree that a geometric model can be used here? Explain your answer.
(3 marks)

(ii) Using an appropriate model, determine the probability that the player will win the game.
(3 marks)

(d) In another game, a player must randomly select 5 balls. If more than 2 balls are lighter than 12 g, then he wins $5. Calculate how much the banker has to charge per game in order to break even.

(7 marks)

(e) Another player draws balls from the box until he obtains one which weights more than 15g. Compute the probability that he does not obtain a ball with more than 15g after drawing 3 balls.

(5 marks)