Question 1

Alice plays 10 rounds of a card game where the outcome of each round is independent of all the other rounds. For each round she can win $8 with a probability of 10% and lose $1 with 90% chance. Let 𝑇 denote the number of rounds out of 10 rounds that she won.

(a) Apply a suitable probability model for 𝑇 and specify the probability distribution for 𝑇. Do include all the relevant parameters of the distribution.

(b) For 𝑖 = 1, 2, ⋯ , 10, the amount that Alice wins in round 𝑖 is given by the random variable 𝑋𝑖 where 𝑋𝑖 = {−1 with probability of 90% 8 with probability of 10% Alice’s net winnings for the 10 rounds is defined as 𝑊 = 𝑋1 + 𝑋2 + ⋯ + 𝑋10. What is Alice’s net winnings for the 10 rounds if she wins exactly 𝑇 = 3 rounds?

(c) Express 𝑊 in terms of 𝑇.

(d) Determine the probability that Alice’s net winnings for the 10 rounds is positive. That is, determine 𝑃(𝑊 > 0).

(e) What is the variance of the random variable 𝑊?

Question 2

The waiting time 𝑇 (in minutes) for Bus 33 outside Tiong Bahru Plaza may be modelled by an exponential distribution with parameter 𝜆 = 0.1.

(a) Calculate the expected value of 𝑇. That is, determine the value of 𝐸(𝑇).

(b) Calculate the median value of 𝑇.

(c) Compute the probability that the waiting time is at most 15 minutes. That is, determine 𝑃(𝑇 ≤ 15).

You arrived at the bus stop outside Tiong Bahru Plaza at 11:15 a.m. and bumped into your friend Ali who was waiting for Bus 33. Ali arrived at the bus stop at 11:00 a.m. and since then no Bus 33 had turned up.

(d) What is the probability that Bus 33 will arrive by 11:30 a.m. or later given that Ali had waited for the bus from 11:00 a.m. to 11:15 a.m.?

(e) Ali believed that the chance that Bus 33 would arrived in the next 15 minutes (from 11:15 to 11:30 a.m.) would be much higher than the answer in part (c) since he had already waited for 15 minutes. Do you agree with his assessment? Justify your answer.

Question 3

In an art class there are 𝐵 = 10 boys and 𝐺 = 15 girls. A group of 3 students is chosen at random from the class. Let 𝑋 denote the number of boys in the group of 3 students chosen at random.

(a) What is the probability that there are more boys than girls in the group of 3 students chosen? That is, determine 𝑃(𝑋 ≥ 2).

(b) Your friend John learned the Binomial distribution in a YouTube video and thinks that it can be applied to obtain a good answer to the question in part (a). John models the distribution of 𝑋 using the Binomial distribution 𝐵(𝑛, 𝑝). Write down the values of the parameters 𝑛 and 𝑝?

(c) Assuming that 𝑋 ~ 𝐵(𝑛, 𝑝) where 𝑛 and 𝑝 have the values stated in part (b), compute 𝑃(𝑋 ≥ 2).

(d) Comment on the answers obtained in parts (a) and (c). Which of the numbers, if any, gives the correct probability of the event that at least two boys are in the group of 3 students selected at random?

Question 4

For this question, you may use the functions in Excel or R. Use of R is optional. Please write down or include all the commands in Excel or R that you use as well as the output.

Import the file UOB.xlsx into the R Studio platform. It contains the daily return on the UOB share over a six-month period from 15 Oct 2021 to 14 April 2022 in percentages. For example, on 15 Oct 2021, the return is 0.41% and so forth. In the following exercises, write down or include all the commands in R that you use as well as the output.

(a) How many data points or observations are there in the six-month data?

(b) Use R to determine the maximum, minimum, median and average return over the six-month period.

(c) Use R to determine the variance and standard deviation of the data.
Is the standard deviation you obtained the sample standard deviation or the standard deviation for a population? That is, does your calculation or command treat the set of data as a sample or as an entire population?

(d) Use R to obtain a histogram of the data. What is the size of each class in the histogram? What does the shape of the histogram tell us about the data?

(e) Draw a random sample of 30 observations from the UOB data set and display the data. Does your sample look similar to the data set? Comment on the sample mean and sample standard deviation in comparison with the entire data set.

Question 5

There are three important components, namely an abductor, a bolt and a coil in a machine. Let 𝐴, 𝐵 and 𝐶 denote the events that the abductor, bolt and coil fail on a given day, respectively. The events 𝐴, 𝐵 and 𝐶 are independent and their probabilities of occurring are 𝑃(𝐴) = 𝑃(𝐵) = 0.2 and 𝑃(𝐶) = 0.3. The machine can continue to function when at most one of the components fail. That is, the machine will fail only when two, or all three, of the components fail.

(a) Compute the probabilities of the events 𝐴 ∪ 𝐵 and 𝐴 ∩ 𝐶. That is, compute 𝑃(𝐴 ∪ 𝐵) and 𝑃(𝐴 ∩ 𝐶).

(b) Show that the events (𝐴 ∪ 𝐵) and (𝐴 ∩ 𝐶) are dependent.

(c) What is the probability that the coil component failed given that the machine failed? That is, compute the conditional probability, 𝑃(𝐶 | machine failed).

(d) What is the probability that the bolt failed on a day that the machine did not fail? That is, compute 𝑃(𝐵 | machine did not fail).