Problem 1

The wait time (after a scheduled arrival time) in minutes for a train to arrive is Uniformly distributed over the interval You observe the wait time for the next trains to arrive. Assume wait times are independent.

(a)

What is the approximate probability (to 2 decimal places) that the sum of the wait times you observed is between and ?

(b)

What is the approximate probability (to 2 decimal places) that the average of the wait times exceeds minutes?

(c)

Find the probability (to 2 decimal places) that or more of the wait times exceed minute.

(d)

Part d) Use the Normal approximation to the Binomial distribution (with continuity correction) to find the probability (to 2 decimal places) that or more of the wait times recorded exceed minutes.

Problem 2

An exam consists of 44 multiple-choice questions. Each question has a choice of five answers, only one of which is correct. For each correct answer, a candidate gets 1 mark, and no penalty is applied for getting an incorrect answer. A particular candidate answers each question purely by guess-work.

Using Normal approximation to Binomial distribution with continuity correction, what is the estimated probability this student obtains a score greater than or equal to 10?Please use R to obtain probabilities and keep at least 6 decimal places in intermediate steps.

# Parameters
n <- 44
p <- 1/5

# Mean and standard deviation
mu <- n * p
sigma <- sqrt(n * p * (1 - p))

# Continuity correction
x <- 9.5  # for P(X >= 10)

# Z-score
z <- (x - mu) / sigma

# Probability
prob <- 1 - pnorm(z)
prob

• [ ] A. 0.5396
• [x] B. 0.3960
• [ ] C. 0.4323
• [ ] D. 0.6040
• [ ] E. 0.2609

Problem 3

65% of the employees in a specialized department of a large software firm are computer science graduates. A project team is made up of 8 employees.

(a)

What is the probability to 3 decimal digits that all the project team members are computer science graduates?

(b)

What is the probability to 3 decimal digits that exactly 3 of the project team members are computer science graduates?

(c)

What is the most likely number of computer science graduates among the 8 project team members? Your answer should be an integer. If there are two possible answers, please select the smaller of the two integers.

(d)

There are 47 such projects running at the same time and each project team consists of 8 employees as described. On how many of the 47 project teams do you expect there to be exactly 3 computer science graduates? Give your answer to 1 decimal place.

(e)

I meet 50 employees at random. What is the probability that the first employee I meet is the first one who is a computer science graduate? Give your answer to 3 decimal places.

(f)

I meet 51 employees at random on a daily basis. What is the mean number of computer science graduates among the 51 that I meet? Give your answer to one decimal place.