Deck 17: When Intuition Differs From Relative Frequency

Question 1

In which of the following situations would the gambler's fallacy not apply?

A)When the events are not independent.
B)When knowledge of one outcome affects the probability of the next one.
C)Both a) and b).
D)Neither a) nor b); the gambler's fallacy holds whenever gambling is going on.That's how casinos stay in business.

Answer 1

Both a) and b).

Question 2

What can happen when a physical situation lends itself to computing a relative-frequency probability, but people ignore that information and go with their 'intuition'?

Answer 2

this can lead to incorrect thinking and to incorrect probabilities, or to incorrect interpretation of probabilities.

Question 3

What is meant by the 'sensitivity' of a test for a certain disease?

Answer 3

The proportion of people who correctly test positive when they actually have the disease.

Question 4

What two conditions cause a test to have a relatively high rate of false positives?

Question 5

Which of the following sequences of six tosses of a fair coin is more likely (if either): HTHTTH, or HHHTTT? Explain your answer.

Question 6

Each possible six-number combination of lottery numbers has an equal (and very unlikely) chance of winning.However, there is a possible edge you can get if you think about people's betting behaviors and the chances of having to split the money or not, if you do pick the winning combination.Give an example of a set of numbers that might give you an edge in that case.

Question 7

In what sense is a coincidence an improbable event?

Question 8

Explain what it means for a test result to be a false positive.

Question 9

Do you agree with the following statement regarding sequences of tosses of a fair coin? Explain why or why not."The sequence HHHHTH doesn't represent the fairness of the coin, and therefore is not as likely to occur as sequences such as HTHTTH."

Question 10

What is meant by the 'specificity' of a test for a certain disease?

Question 11

Which of the following beliefs are examples of the gambler's fallacy?

A)Random events are self-correcting.(For example if you are losing, believing that your luck is about to turn around.)
B)The long-run frequency of an event should apply in the short term as well.(For example, "He's a 90% free throw shooter, and he's already missed 2 out of 3 tonight.There's no way he's going to miss this next one.")
C)Knowledge of one event will help predict the next event, even though the events are independent.(For example, you may hear someone at the craps table saying to the person rolling the dice, "You're on a roll! Don't stop now!")
D)All of the above.

Question 12

When you experience a coincidence, which of the following interpretations is appropriate?

A)If an event has a million to one chance, it is expected to happen to 290 people in the U.S.in a given day, on average (because the U.S.population is 290 million).
B)It is not unlikely that something surprising will happen to someone, somewhere, someday.
C)There is a big difference between the probability of a rare event happening to someone somewhere, and the probability of a rare event happening to a specifically named individual.
D)All of the above.

Question 13

Which of the following lottery combinations (6 numbers from 1-30, no repeats allowed) is the least likely to come up as a winning ticket?

A)2, 10, 23, 27, 30, 11
B)1, 2, 3, 4, 5, 6
C)11, 19, 14, 12, 17, 15
D)All of the above are equally unlikely.

Question 14

The probability of a coincidence happening may be __________ than you think.

Question 15

About how many people would need to be gathered together to be at least 50% sure that two of them will share the same birthday (the same day of the year, not necessarily the same year)?

A)4
B)23
C)183
D)2,300 or more

Question 16

A(n) __________ is a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection.

Question 17

Explain how a misinterpretation of a coincidence could impact a town that is experiencing an unusually high rate of cancer, compared to the rest of the country.

Question 18

Which of the following coincidences reflects a truly improbable event, if interpreted properly?

A)My next-door neighbor won the lottery! I was only one house away from becoming rich!
B)Wow, what are the chances that I'd be on the same flight as Bill Cosby? Unbelievable!
C)I heard there's a guy in Pennsylvania who won the lottery twice; that lottery must be rigged.
D)None of the above.

Question 19

How does the relative frequency interpretation of probability differ from the personal probability interpretation, in terms of variability in the results?

Question 20

In what sense is a coincidence not an improbable event?

Question 21

The __________ of a test for a certain disease is the proportion of people who correctly test positive when they actually have the disease.

Question 22

Which of the following describes the 'specificity' of a test for a certain disease?

A)The probability that you are likely to have a specific disease, without any knowledge of your test results.
B)The proportion of people who correctly test positive when they actually have the disease.
C)The proportion of people who correctly test negative when they don't have the disease.
D)All of the above.

Question 23

Which probability is larger for a randomly selected woman from the general population: The probability of having breast cancer given that the woman has a positive mammogram, or the probability of a positive mammogram given that the woman has breast cancer?

A)The probability of having breast cancer given that the woman has a positive mammogram.
B)The probability of a positive mammogram given that the woman has breast cancer.
C)The probabilities are the same.
D)Not enough information to tell.

Question 24

Research shows that people tend to "detect" patterns even where none exist, and to __________ the degree of clustering of outcomes in sports events, as in other sequential data (for example the chance of being on a shooting streak in basketball).

Question 25

Suppose a lottery game has a 1/1000 chance of winning for each ticket, and the prize is $1,000.How much should you be willing to pay for a ticket in order to have a positive expected value?

Question 26

Which of the following would be true if people made decisions based on maximizing their expected monetary return?

A)People wouldn't buy lottery tickets.
B)People wouldn't buy insurance.
C)People wouldn't participate in sports betting.
D)All of the above.

Question 27

Which of the following problems can arise when intuition differs from relative frequency?

A)Always thinking that a coincidence has a low chance of occurring.
B)The gambler's fallacy.
C)Confusion of the inverse.
D)All of the above.

Question 28

What is meant by a test result that is a false positive?

A)You have the disease but the test results were negative.
B)You don't have the disease but the test results were positive.
C)You test positive for a different disease than the one they thought you had.
D)None of the above.

Question 29

Which of the following is needed in order to determine the probability of a positive test result being accurate?

A)The base rate.
B)The sensitivity of the test.
C)The specificity of the test.
D)All of the above.

Question 30

If the bookmakers who run sports betting operations were to set fair odds, so that both the house and those placing bets had expected values of zero, then the probabilities for all of the players should sum to 1, but they don't.Do they sum to a higher or lower value, and why?