DS280 - Intro to Statistics

Homework #2 - Probability II

The purpose of this assignment is to give you further experience with probability problems that occur in the “real world.” The assignment is worth 40 points (the equivalent of two quizzes). It is due at the beginning of class on Wednesday, February 3. Recall that writing “Pledged” before your signature on the submitted assignment is a reminder of your ongoing commitment to the Honor System. You are welcome (and encouraged) to work with others in the class on these problems. However, the write-up must be your own, in your own words.

Remember that professionalism matters in this class. While this assignment need not be typed, it should be exceedingly legible. Each problem should be on a separate sheet of paper. (You may not need to fill the page. However, this will make grading easier.) You will of course staple the pages together. If I can't read it or find it, I can't give you credit for it. Several of the problems are challenging, so you are advised not to wait to the last minute to begin the assignment.

Question 1 [5 points]:

Oblivion is a computer-based role-playing game. Players create an imaginary character then live out that character’s adventures as he/she goes about basically saving the world and other simple tasks.

A wide range of options is available in creating this fictional alter ego. You may choose the character’s gender (male or female, of course), race (ten different selections ranging from humans to elves to orcs to sentient catlike or lizardlike creatures), specialization (combat, magic, or stealth), principal attributes (select two from a list of eight – things like intelligence or strength or personality), major skills (primary abilities – select seven of twenty-one options, ranging from swordsmanship to alchemy to sneakiness), and birthsign (the Morrowind zodiac has thirteen signs).

How many different characters could be created?

Question 2 [10 points]:

Remember that a standard deck of cards contains four suits (clubs, diamonds, hearts, spades) with thirteen cards in each (Ace, King, Queen, Jack, Ten, Nine, Eight, Seven, Six, Five, Four, Three, Two), for a total of 52 cards.

In straight poker, each player is dealt five cards. (Other forms of poker allow for drawing, discards, additional cards, wild cards, etc. We’ll ignore these complications for now, and play the game in its purest form.)

a) What is the probability that you are dealt a flush – that is, all five cards are from the same suit?

a) What is the probability that you are dealt four of a kind – that is, all four cards of a given rank? (For example: all four Aces, or all four Sevens.)

Question 3 [10 points]:

Medical technology has developed numerous procedures for testing for various diseases, drug use, etc. All such procedures are prone to some error. There is therefore a real concern with the occurrence of “false positives” (saying a subject has the condition when he in fact doesn't) and “false negatives” (saying the subject does not have the condition, when he actually does).

Suppose that a decision is made to test all driver's license applicants for drug use (or: all college athletes for drug or steroid use; or: all prison inmates for AIDS). Let's consider what the false positive rate might be, under a variety of circumstances.

a) Suppose first that everyone is tested for the condition, and that in reality two percent of the population has the condition. Suppose also that the test procedure used has 95% reliability - that is, whatever your condition, the test evaluates it correctly 95% of the time. What is the false positive rate? The false negative rate?

b) One way to lower the false positive rate is to modify the test in a way that will make it harder to get a “positive” reading, and easier to get a “negative” one. This will affect the reliability of the instrument. Let's suppose such a modification is made, which increases to 97.5% the chance of correctly identifying a person without the condition, but which lowers to 90% the chance of correctly identifying a person with the condition. Now, what is the false positive rate for the problem in Part A? The false negative rate?

c) Another way to lower the false positive rate is only to test some members of the population - those deemed particularly “at risk,” or for whom there is some “probable cause” to suspect presence of the condition. Let's suppose such a preliminary screening is done, eliminating much of the population from consideration. Of the remaining group, on whom the test is done, suppose that fully one-half have the condition of interest. Repeat Part A. Now, what are the false positive and false negative rates?

d) What are the relative costs of false positives and false negatives? What implications do these results have for various testing programs? (You may wish to consider things in context of a particular situation. Feel free to bring in outside information.)

Question 4 [15 points]:

In February, 1986, Evelyn Adams of New Jersey garnered national headlines when she won the state lottery - for the second time. This problem looks at just how unusual it is for someone to win two state lotteries.

a) A New Jersey State Lottery ticket contains 42 numbers; the bettor selects six of these. Six numbers are drawn by the lottery commission; if the bettor matches all six numbers, he/she wins the grand prize. If you buy a single lottery ticket, what is the probability that you win the grand prize? If you buy 20 tickets one week [with different selections of numbers, of course], what is the probability that you win the grand prize?

b) The newspaper story reported that Adams purchased 20 tickets a week. Suppose she were to do so for 30 years. What is the probability that she wins the lottery twice during those 30 years?

c) Suppose 200,000 people are purchasing lottery tickets with the same frequency as Adams. (Note that this number is simply a guesstimate - but is quite probably too low. There are a lot of hardcore lottery players out there.) What is the probability that at least one of them would win two lotteries?