Likelihood and Uncertainty: Understanding Probabilities

Chapter 7 Likelihood and Uncertainty: Understanding Probabilities Active Learning Exercises

Practice the likelihood and uncertainty critical thinking skills that you've learned in this chapter.

Exercise 7.1

How many possible outcomes are there in four flips of a fair coin? What is the probability of any particular outcome? Why would a guess of “H-T-H-T” be likely to be wrong? Why would a guess of “about half heads and half tails” be likely to be correct?

Exercise 7.2

In Sue’s classroom, they played a game, where all the students (about 35 of them) announced their birthdays out loud. Two people had the same birthdate as Sue! Explain this shared birthday in terms of probabilities and likelihoods.

Exercise 7.3

Tell a friend about the event described above. Ask him or her to explain the event. Is their explanation different from yours? In what ways?

Exercise 7.4

Poll your friends (or your friends’ parents or parents’ friends) regarding their preferred games of chance in Las Vegas (and elsewhere). Ask them why they have such preferences. Do they follow any “luck” rituals (e.g., lucky numbers, carrying a luck charm, etc.)? Try to explain their responses.

Exercise 7.5

Poll your friends about their “big wins” or big losses in Las Vegas (or when playing any game of chance, including state lotteries). How do these events affect their game preferences? How do your friends explain those events?

Exercise 7.6

In 1997, Noyes hospital in Dansville, NY reported a string of 12-straight girls born in a row (Livingston County (NY) News, August 1997). If a birth rate of 50% boys-50% girls is normal, what is the probability of such an event? Of the total birth rate in the hospital in 1997, 80 of 168 total births were girls. What is the probability of this event? What does this example tell us about events as they occur in “the long run?” Exercise 7.7

There is an advertisement that appears on late-night television that offers to sell the secrets to picking winning lottery numbers. The advertisement tells viewers not to play the lottery in a stupid way. Is it possible to play the lottery intelligently by paying for the secret numbers, or is this "bunk?" Explain your answer.

Exercise 7.8

Suppose now that you learn that most people pick important dates in their life when they select lottery numbers (e.g., birthdays, anniversaries). Could you increase the amount of money that you could win by picking numbers that do not correspond to dates such number greater than 31? Would this system make you more likely to win? Would this system make it more likely that you would win a big jackpot, if you did win? Explain your reasoning. (I thank Dr. Dale Berger at Claremont Graduate School for suggesting this problem.)

Exercise 7.9

If each of the letters in the word "PROBABILITY" were thrown separately into a hat and one letter is drawn from the hat, what is the probability that it is a vowel? ("Y" is a vowel in this example.)

Exercise 7.10

Your friend is willing to give you 5:2 odds that the Phillies will beat the Dodgers. Convert these odds to a probability value.

Exercise 7.11 a) What is the probability of drawing a picture card (Jack, Queen, or King) from a full deck of 52 cards? b) What is the probability of drawing two aces in a row from a full deck (without replacement)?

Exercise 7.12

In a party game called "Spin the Bottle," the players form a circle with a bottle at the center. A spinner spins the bottle and then kisses the person to whom it points. Although five people are playing, the bottle has pointed to Marlene on each of its three spins. What is the probability of this occurring by chance? Can you make any "guesses" about Marlene? (hint: try a tree diagram)

Exercise 7.13 Professor Aardvark gives such difficult exams that students can only guess at the answers. a) What is the probability of getting all 5 questions on his True/False test correct by guessing (assuming that every alternative is equally likely to be correct)? b) What is the probability of getting all 5 wrong?

Exercise 7.14

Every student in Mr. Weasel's class kept a record of the number of books read over a 3-month period. The data are: 15, 5, 8, 12, 1, 3, 1, 7, 21, 4. Compute the mean and median for these numbers. Which one seems better as a measure of central tendency?

Exercise 7.15

Rubinstein and Pfeiffer (1980) have suggested that instead of reporting weather forecasts in terms of the probability of rain, a more useful index would be the Expected Value (EV) of rain. Suppose the weather forecaster knows that there is a 30% probability of 5 inches of rain and a 70% probability of no rain on a given day. What is the EV for rain? Is this number more useful than the probability of rain?

Exercise 7.16

Abby scored very low on her SAT's. Her score of 350 is well below the mean of 500. If she retakes the test, which of the following is most likely to occur and why did you pick that answer?

a. She will probably score below 350. b. She will probably score close to 350. c. She will probably score between 350 and 500. d. She will probably score near 500. e. She will probably score above 800.

Exercise 7.17

A gourmet critic found that she is frequently disappointed when she returns to restaurants that she found to be outstanding on her first visit. She concludes that the chefs get lazy over time and do not put as much effort into their cooking. What do you think about her reasoning?

Exercise 7.18

Some people believe in what is called the “Sports Illustrated Curse,” where an athlete who appears on the cover of Sports Illustrated is subsequently “cursed”, so his or her previous star performance declines. What is a more reasonable explanation of this “curse?”

Exercise 7.19 When the College Board announced that, on average, males score somewhat higher on the mathematics portion of the test, Darryl protested. His sister scored a perfect 800 on this test and his girlfriend scored "way better" than he did. Comment of Darryl's protest.

Exercise 7.20

You joined an office "pool." There are 8 teams that you can bet on. You bet on the mighty Coyotes that have won 40% if the games played so far this season. You decided to bet $5.00 on them. If you lose, you lose your $5.00. If you win, you get back your $5.00 and get another $5.00. If you bet on the Coyotes all season and the probabilities remain the same, what is the expected value of each bet? How much should you expect to win or lose if you bet on 10 games?

Exercise 7.21

You are planning on a beautiful picnic. You will only go on your picnic if it doesn't rain, and your friend goes with you, and the park is open. The mean old weather forecaster is predicting a 40% chance of rain. Not only that, but your friend is only 75% sure that he will go with you. The park ranger is 90% sure that the park will be open this time of year. What is the probability that it won't rain, and your friend will go with you, and the park will be open (assuming independence)? Before you start this problem, estimate your answer.

Exercise 7.22

How certain can we be about the occurrence of an event if its probability is 0? .2? .5? .9? 1.0?

Exercise 7.23

Two sports fans are arguing over which sport – baseball or football – has the best (most accurate) playoff system. Charlie says that the Super Bowl is the best way of determining the world champion because, according to him, “the seven games of the World Series are all played in the home cities of the two teams, whereas the Super Bowl is usually played in a neutral city. Because you want all factors not related to the game to be equal for a championship, then the Super Bowl is the better way to determine the world champion” (Jepson, Krantz, & Nisbett, 1983). Which is a more accurate measurement of the true championship team – World Series or Super Bowl? Why?

Exercise 7.24

You were just thinking about an old friend from high school when your mother walks into the room and tells you that she bumped into this old friend. Isn't this amazing? Don't you think that you can use this example to convince your skeptical friends that you have ESP? Explain your answer.

Exercise 7.25 Ask several friends to remember something (an event, a song title, an important historical date, etc.—something they may have some difficulty remembering). After an interval of approximately one week, ask your friends to recall the information. Ask them to rate how confident they are in their accuracy (on a scale of 1 to 10, with 1=not at all confident and 10=completely confident). Are they all very confident? Are they all confident, even if they are inaccurate?

Exercise 7.26

Ask two friends to play a card game. Allow one friend to select his or her own card from a “face-down” deck; you choose a card and give it to the other person. Without looking at the cards, ask them to rate their confidence (on a scale of 1 to 10) that they have the high card. Repeat this several times, without showing your friends how had the higher card on each trial, always allowing one person and not the other to choose their card from the deck. Do you expect the person who selects his or her own card to give different answers than the person who is given a card? Is one person consistently more confident that they have the high card? Explain why or why not.

Exercise 7.27

The Bible Code (Drosnin, 1997) claims that hidden codes in the Bible reveal events that occurred after the Bible was written, including contemporary events. The codes usually involve letters that are equidistant from each other throughout passages of text, not counting punctuation or spacing. Read the following Bible passage (taken from Thomas, November 1997, Skeptical Enquirer):

“And hast not suffered me to kiss my sons and my daughters? Thou hast now done foolishly in so doing.” (Genesis 31:28)

Use the following code (similar to those used by Drosnin) to find the “hidden” message. Start at the R in “daughters,” skip over three letters (not counting punctuation) to the O in “thou”, and three more to the S in “hast” and so on. What is the message? Do you think the writer(s) of Genesis had a psychic premonition of this event that they were trying to communicate to readers of the Bible?

Exercise 7.28

Officials at the suicide prevention center know that 2% of all people who phone their hotline actually attempt suicide. A psychologist has devised a test to help identify those callers who will actually attempt suicide. She found that 80% of the people who will attempt suicide attain a positive score on this test, but only 5% of those who will not attempt suicide attain a positive score on this test. If you get a positive identification from a caller on this test, what is the probability that he would actually attempt suicide?

Hint: Set up a matrix like the one shown below and then fill in the appropriate numbers. Score Will Attempt Suicide Will Not Attempt Suicide Row Total Positive Score Negative Score Column Totals

Draw a tree diagram with 4 branches, multiply along the branches, then form the correct ratio. The numerator will be the proportion that would actually attempt suicide and the test predicts that they will over this number plus the proportion that do not attempt suicide and the test predicts that they will. Follow the example in the book for José, if you're having trouble getting started.

Exercise 7.29

A conditional probability is the probability that an event will occur given that another event has already occurred. Unlike a joint probability (the probability of two independent events occurring together), a conditional probability involves dependency between the events. Of the following events, does the probability of one event change if the other has already occurred? Why or why not?

a. drinking alcohol and getting into a car accident b. physical fitness and health problems c. heads on the first flip of a coin and heads on the second flip of a coin d. smoking cigarettes and developing lung cancer e. licensing your dog and your dog being lost and not returned to you f. the probability of an unplanned pregnancy and regular use of contraceptives

Exercise 7.30

A formula called Bayes Theorem is used to determine conditional probabilities (shown below), where p(A/B) equals “the probability of event A given that event B has already occurred;” p(B/Ā) equals “the probability of event B given that event A has not occurred;” and p(Ā) is the probability of event A not occurring and p(A) is the probability of event A.

p(A/B)= p(B/A) p(A) p(B/A) p(A) + p(B/Ā) p(Ā)

Using Bayes Theorem, consider the following: The standard test for the HIV virus has a 99% sensitivity rate and a 99% specificity rate. 99% sensitivity means that for every 1000 people tested who do have HIV, we can expect 990 to test positive and 10 to have a false negative test (a negative test even though they actually do have HIV). 99% specificity means that for every 1000 people who do not have HIV, we can expect 990 to get an accurate negative test result and 10 to have a false positive test result (a positive test even though HIV is not present). Based on this scenario, if only 5% of the population is actually HIV positive, what is the probability that you are HIV positive given a positive test result (p(A/B)? Before you work through the arithmetic, first estimate the correct answer. After solving for the answer, think about it. Was it (much) higher or lower than your original estimate? If so, explain why.

Exercise 7.31

Using the information above, how would you react to a friend who just told you that he received a positive HIV test result, suggesting he is HIV positive? What suggestions would you make to reduce his anxiety?

Exercise 7.32

In the emergency stay order issued by the U.S. Supreme Court that halted the recounting of undervotes in Florida in the 2000 Presidential Election, Justice Antonin Scalia wrote that continuing the recounting of votes not counted because of machine error in all counties in Florida would “threaten irreparable harm to George Bush” (Bugliosi, 2001). Why was Justice Scalia wrong if the votes not counted because of machine error occurred randomly (i.e., in equal frequency for George Bush and Al Gore, based on the proportion of voters selecting each)? If the undervotes occurred randomly, did stopping the recount threaten irreparable harm to Al Gore? Why?

Exercise 7.33

When Supreme Court Justice Antonin Scalia ordered a halt to the recounting of 60,000 undervotes in Florida in the 2000 Presidential Election, the difference between George Bush and Al Gore was 154 votes (Bugliosi, 2001). Based on the long run and the laws of large and small numbers, would counting 60,000 votes likely change a 154-vote difference? What are the implications of this reasoning, and what does it suggest about the validity of the emergency stay order?

Exercise 7.34

A newspaper article (LA Times, 2001, November 12) suggested that of the undervotes in Florida in the 2000 Presidential Election, black voters votes’ were more likely to not be counted because of machine error. The authors argued that black Republicans were especially likely to have their votes not counted. Comment on this argument based on your knowledge of politics and probability. Are the authors justified in their conclusion that Republican voters were less likely than Democrats to have their votes counted, if machine errors occurred at random?

Exercise 7.35

After the World Trade Center was attacked on September 11, 2001, a news program reported a story entitled, “The Miracle of Ladder Company 6,” about a small group of firefighters who survived the collapse of one tower because they were in a reinforced stairwell. Comment on the “miraculous” nature of this event. How does it reflect a search for meaning? Are there other explanations for this event, besides a miracle? Does use of the word “miracle” affect one’s evaluation of the event? How so?