MathStat474 - HW 3

Posted: Tue Jan 27. Due: Thu Feb 5th.

Problem 1

Solve exercise 3.11.3 in the textbook. (Formulate a sample space.)

Provide a reasonable description of the sample space for the following random experiment: In the final inspection of electronic power supplies, either units pass or three types of nonconformities might occur: functional, minor, or cosmetic. Three units are inspected.

Problem 2.

Solve exercise 3.11.15 in the textbook. (Sample space, events, experiments, and illustration using a Venn diagram.)

A digital scale that provides weights to the nearest gram is used. a) What is the sample space for this experiment? Let A denote the event that a weight exceeds 11 grams, let B denote the event that a weight is less than or equal to 15 grams, and let C denote the event that a weight is greater than or equal to 8 grams and less than 12 grams. Describe the following events:

  1. \(A\cup B\)
  2. \(A\cap B\)
  3. \(A'\) (complement of \(A\))
  4. \(A\cup B\cup C\)
  5. \((A\cup C)'\) (complement of this union)
  6. \(A\cap B\cap C\)
  7. \(B'\cap C\)
  8. \(A\cup (B\cap C)\).

Problem 3.

Exercise 3.11.25 in the textbook: computing the number of outcomes (in a given sample) in each of the events.

The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of Chronic Hepatitis C” [Gastroenterology (1996, Vol. 111(5), pp. 1307–1312)], considered the effect of two treatments and a control for treatment of hepatitis C. The following table provides the total patients in each group and the number that showed a complete (positive) response after 24 weeks of treatment.

Treatment Complete Response Total
Ribavirin plus interferon alfa 16 21
Interferon alfa 6 19

Let A denote the event that the patient was treated with ribavirin plus interferon alfa, and let B denote the event that the response was complete. Determine the number of patients in each of the following events.

  1. \(A\)
  2. \(A\cap B\)
  3. \(A\cup B\)
  4. \(A'\cap B'\)

Problem 4.

Solve exercise 3.11.35 in the book. (Counting and computing probabilities with equally likely outcomes in the sample space.)

In the laboratory analysis of samples from a chemical process, five samples from the process are analyzed daily. In addition, a control sample is analyzed twice each day to check the calibration of the laboratory instruments.

  1. How many different sequences of process and control samples are possible each day? Assume that the five process samples are considered identical and that the two control samples are considered identical.
  2. How many different sequences of process and control samples are possible if we consider the five process samples to be different and the two control samples to be identical?
  3. For the same situation as part (b), how many sequences are possible if the first test of each day must be a control sample?

Problem 5.

Solve exercise 3.11.65 in the textbook. (Computations of probabilities, using the first 5 rules of probability.)

A computer system uses passwords that are six characters, and each character is one of the 26 letters (a–z) or 10 integers (0–9). Uppercase letters are not used. Let \(A\) denote the event that a password begins with a vowel (either a, e, i, o, or u), and let \(B\) denote the event that a password ends with an even number (either 0, 2, 4, 6, or 8). Suppose a hacker selects a password at random. Determine the following probabilities:

  1. \(P(A)\)
  2. \(P(B)\)
  3. \(P(A\cap B)\)
  4. \(P(A\cup B)\).

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