ENGC 6310

Assignment 2

Submission of the assignment is one week after the end of Lecture 4.

Problem 1:

Problem 2:

In a company that manufactures computer chips, three chips from a pool of chips are tested. Each

chip is classified as pass or fail. Assume that the probability that a chip does not pass the test is

0.2 and that chips are independent, determine the probability mass function, the mean, and the

variance of the number of chips that pass the test.

Solve by Binomial PDF

Problem 3:

In a company that manufactures computer chips, fifty chips from a pool of chips are tested. Each

chip is classified as pass or fail. Assume that the probability that a chip does not pass the test is

0.2 and that chips are independent.

1. Determine and draw the probability mass function (PMF), the mean, and the variance

of the number of chips that pass the test (Presenting PMF using MS-Excel).

2. Determine and draw the probability mass function, the mean, and the variance of the

number of chips that do not pass the test (Presenting PMF using MS-Excel)

3. Use the normal approximation to evaluate the probability that 20 defective chips or

less are found. Compare this probability with the probability value obtained in part

“a”

Solve by Binomial PDF

Problem 4:

A particularly long traffic light on your morning commute is green 25% of the time that you

approach it. Assume that each morning represents an independent trial.

a) Over six mornings, what is the probability that the light is green on exactly one day?

b) Over 20 mornings, what is the probability that the light is green on exactly three

days?

c) Over 20 mornings, what is the probability that the light is green on more than four

days?

d) What is the probability that the first morning that the light is green is the fourth

morning that you approach it?

e) What is the probability that the light is not green for 10 consecutive mornings?

Problem 5:

In a clinical test of a specific medical drug, it was found that 6% of people who take the drug

experienced headaches.

1. Assuming that the 6% rate applies, find the probability that among eight drug users,

three experience headaches.

2. Assuming that the 6% rate applies, find the probability that among eight drug users, all

eight experienced headaches.

Solve by Binomial PDF

Problem 6:

Use MS-Excel to draw the following pmf’s:

1. Poisson distribution with λ = 0.5

2. Poisson distribution with λ = 5 (use MS-Excel built-in function of Poisson distribution)

3. Poisson distribution with λ = 10 (use MS-Excel built-in function of Poisson

distribution)

Problem 7:

The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson

distribution with a mean of 0.05 flaw per square foot of plastic panel. Assume an automobile

interior contains 10 square feet of plastic panel.

a) What is the probability that there are no surface flaws in an auto’s interior?

b) If 10 cars are sold to a rental company, what is the probability that none of the 10

cars has any surface flaws?

c) If 10 cars are sold to a rental company, what is the probability that at most one car

has any surface flaws?

Problem 8:

The Environmental Protection Agency used a tailpipe test to determine which of 116,667 cars

generated too much pollution. It is estimated that 1% of cars fail such a test.

a) If we randomly select 20 cars from the group of 116,667, how many are expected to

fail the tailpipe test?

b) Find the mean and standard deviation for the numbers of cars in groups of 20 that fail

the tailpipe test.

c) Find the probability that in a randomly selected group of 20 cars, there is at least one

that fails the tailpipe test.

d) Is it unusual to find that in a group of 20 randomly selected cars, there are 3 that fail

the tailpipe test? Why or why not?

e) If two different cars are randomly selected, find the probability that they both fail the

tailpipe test.

Problem 9:

Suppose that the concrete slab breaking strengths measured in certain units are between 120 and

150, with a probability density function of:

a) Sketch the pdf

b) Determine and sketch the cdf

c) Calculate the probability that the breaking strength of a randomly selected

slab is below 130.

d) Calculate the probability that the breaking strength of a randomly selected

slab is between 130 and 145.

e) Calculate the mean (expectation) of the random variable X

f) Calculate the variance of the random variable X

g) Calculate the median of the random variable X.

h) What value of breaking strength that is exceeded by 85% of the samples?

i) Find the interval in which we can be 95% confident that a randomly selected

concrete slab has a breaking strength within this interval.

Problem 10:

Assume X is normally distributed with a mean of 5 and a standard deviation of 4.

Determine the value for x that solves each of the following:

(a) P(X > x) = 0.5 (b) P(X > x) = 0.95

(c) P(x < X < 9) = 0.2 (d) P(3 < X < x) = 0.95

(e) P(– x < X < x) = 0.99

f) Present the probability mass functions by MS-Excle.

Problem 11:

The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean

of 10 minutes.

a) What is the probability that you wait longer than one hour for a taxi?

b) Suppose you have already been waiting for one hour for a taxi, what is the probability

that one arrives within the next 10 minutes?

c) Present the probability mas function by MS-Excel.

Problem 12:

Weights of individual concrete blocks manufactured by a company are known to be normally

distributed with a mean of μ = 11 kg and a standard deviation of σ = 0.3 kg.

a) Find the probability that a concrete block weighs less than 10.5 kg.

b) Find the interval in which the company can be 99% confident that a randomly selected

concrete block has a weight within this interval.

c) Find the interval in which the company can be 99% confident that the mean weight of 20

randomly selected blocks is within this interval.