Deck 14: Understanding Probability and Long-Term Expectations

Question 1

Suppose Bob and Mary are each trying to determine whether or not a coin is fair (that is, the coin has an equal chance of landing heads up or tails up).Bob flips the coin 100 times and finds that it lands heads up 80 times.Mary flips the coin 1,000 times and finds that it lands heads up 800 times.Compare the relative frequencies in each case and discuss who (if anyone) has stronger evidence that the coin is unfair.

Answer 1

mary has stronger evidence.Bob and mary both found a relative frequency of 0.8 or 80%, but since Mary's data is based on many more observations, hers is more likely to be close to the "true probability" than bob's.

Question 2

Which of the following is not an example of using the relative-frequency interpretation of probability?

A)Buying a lottery ticket every week and observing whether it is a winner.
B)Testing individuals in a population and observing whether they carry a gene for a certain disease.
C)Being a member of a jury and deciding that the chance of the defendant being innocent is very small.
D)All of the above are examples of using the relative-frequency interpretation.

Answer 2

Being a member of a jury and deciding that the chance of the defendant being innocent is very small.

Question 3

Suppose you know that for each ticket the probability of winning a certain "instant win" lottery game is 1/100.You have purchased 99 tickets so far, with no luck.What is your chance that the next ticket you buy is a winner?

A)99/100
B)1/100
C)100%
D)None of the above.

Answer 3

1/100

Question 4

Give two examples of ways that we speak about probability in our every day lives.

Question 5

The airlines routinely report their on-time flight percentages, which can be interpreted as probabilities.What method of finding probabilities was most likely used in determining this?

Question 6

Describe a situation where you can't use the relative frequency approach to interpret probability.

Question 7

Which of the following is an example of using the personal probability interpretation of probability?

A)Buying a lottery ticket every week and observing whether it is a winner.
B)Observing the percentage of time that all of the prices ring up correctly when you visit a certain store each week to do your shopping.
C)Deciding that your football team has a high chance of winning their next game.
D)All of the above are examples of using the personal probability interpretation.

Question 8

Name two restrictions on personal probabilities, from a statistical standpoint.

Question 9

Give an example of a situation where you have to use the personal probability interpretation of a probability (vs.the relative frequency interpretation).

Question 10

Suppose Consumer Reports tests a random sample of 1,000 flashlights of a certain brand and finds 20 of them to be defective.They report the chances of buying a defective flashlight (for this brand) to be .02.This is an example of using the _______________ approach to determining a probability.

Question 11

What is one caution you should observe when encountering personal probabilities that are reported in the media?

Question 12

Which of the following is not an example of a statement based on a personal probability?

A)"Based on his credentials and experience, I believe this candidate has a high chance of being successful here."
B)"I read in a study that 98% of the patients who received this vaccine suffered no side effects, so my chances of developing side effects are very small."
C)"We believe beyond a reasonable doubt that the defendant is guilty."
D)All of the above are statements based on personal probability.

Question 13

Tell whether the following statement is correct; if it is not correct, explain the problem."60% of the voters were in favor of Issue 1.That means 40% of them must have opposed it."

Question 14

Tell whether the following statement is correct; if it is not correct, explain the problem."If the probability of a single birth resulting in a boy is .51, then the probability of it resulting in a girl is also .51."

Question 15

Which of the following statements is not true regarding personal probabilities?

A)They often take relative frequencies of similar events into account.
B)They are based on unique situations that are not likely to be repeated.
C)We could each assign a different personal probability to the same event.
D)All of the above are true statements regarding personal probabilities.

Question 16

Which of the following chances can be quantified exactly?

A)The chance that it will rain tomorrow.
B)The chance that you will live to be 100.
C)The chance of getting four of a kind in a poker hand.
D)All of the above.

Question 17

Using the relative frequency approach, we can define the probability of any specific outcome as the __________ of times it occurs over the long run.

Question 18

Suppose you encounter one traffic light on your commute to work each day.You have determined that the probability that this light will be red is 1/3.Which of the following is not a correct interpretation of this probability?

A)The light will always be red one out of every three times that you encounter it.
B)In the long run, the light should be red about 33.33% of the time.
C)Each time you approach the light on your commute, the probability of it being red is 1/3.
D)All of the above are correct interpretations.

Question 19

Give an example where you can use the relative-frequency approach to determine a probability.

Question 20

Which of the following does not apply to the relative-frequency approach for trying to determine a probability of a specific outcome?

A)The probability is determined to be the proportion of times the outcome occurs in the long run.
B)The relative frequency jumps around if only a few observations are made, but eventually settles down to a certain proportion as more and more observations are made.
C)The relative frequency approach to determining a probability can be applied to short-term observation.
D)All of the above statements apply to the relative-frequency approach.

Question 21

Which probability is the smallest?

A)The probability that a couple's third child is a girl.
B)The probability that a couple has their first girl by the time of the third child.
C)The probability that a couple's first girl occurs the third time around.
D)All of the above probabilities are equal.

Question 22

If two events are mutually exclusive, they __________ be independent.

Question 23

For Questions use the following narrative:
Narrative: Grades
Suppose a class of 100 students took their statistics final and their grades are shown in the table below.

{Grades narrative} Choose one student at random.What is the probability that he/she received a B or a C?

Question 24

Suppose 40% of the people in a population smoke 20 cigarettes per day, and the remaining 60% smoked none.Which of the following involves a correct interpretation of the expected value?

A)For all people in the population, the average is 8 cigarettes smoked per person per day.
B)For all people in the population, the average is 20 cigarettes smoked per person per day.
C)The chance is high that the next person you meet smokes eight cigarettes a day.
D)Almost everyone in the population smokes close to eight cigarettes per day.

Question 25

For Questions , use the following narrative:
Narrative: Instant lotto
Suppose an "Instant Lotto" ticket costs $5, and the chances of winning the $500 prize are 1/10,000.There are no other prizes.
{Instant lotto narrative} What is your expected value for this game for each ticket you buy?

Question 26

For Questions , use the following narrative:
Narrative: Instant lotto
Suppose an "Instant Lotto" ticket costs $5, and the chances of winning the $500 prize are 1/10,000.There are no other prizes.
{Instant lotto narrative} Interpret what the expected value would mean in this situation, in words that a non-statistics student would understand.(It is not necessary to calculate the expected value here.)

Question 27

Suppose 5% of the people who buy a certain type of DVD player return it to get their money back.The DVD player costs $100.What is the expected loss, per customer, for the company due to returns (ignoring the monetary value of the returned DVD player)?

A)$5
B)$0; most people don't return it
C)Either $0 or $100, but it is impossible to determine which.
D)$100

Question 28

Which of the following is a true probability?

A)-0.22
B)120%
C)1
D)None of the above

Question 29

Which of the following is an example of a cumulative probability?

A)The chance of being infected with a sexually transmitted disease by the 10th independent encounter.
B)The chance of first becoming infected with a sexually transmitted disease during the 10th encounter.
C)The total percentage of encounters resulting in the spread of a sexually transmitted disease.
D)All of the above.

Question 30

Suppose a slot machine has an expected payout of .95 cents on the dollar, meaning for every dollar you play, you get 95 cents back.What is your expected value for the amount lost per dollar played on this machine?

A)$0.95
B)-$0.05
C)-$1.00
D)The entire jackpot, if you play long enough.

Question 31

Suppose your commute to work involves encountering 3 inter

A)3/10 = .30 or 30%
B)2/100 = .02 or 2%
C)3/20 = .15 or 15%
D)None of the above

Question 32

For Questions use the following narrative:
Narrative: Grades
Suppose a class of 100 students took their statistics final and their grades are shown in the table below.

{Grades narrative} What is the probability that a student selected at random passed the final (where a D is considered to be a passing grade)?

Question 33

For Questions use the following narrative:
Narrative: Shaking hands
Suppose the chances of picking up a cold from someone by shaking hands with them is .01 (assuming you don't know whether they have a cold or not), and that each encounter you have is independent of another.
{Shaking hands narrative}.Suppose you shake hands with 5 people in a given day.What is the probability that you don't pick up a cold from any of these people?

Question 34

Suppose the chance that your plane will depart and arrive on time in the same trip is 90%.The chance that your plane will depart on time on any given trip is __________ 90%.(Choose one: at most, equal to, or at least, and fill in the blank.)

Question 35

Which of the following describes an example of an expected value in a lottery situation?

A)The average amount of money you'll win/lose in the long run when playing the lottery.
B)The amount of money you will win/lose on any given ticket.
C)The amount of money that you'll have the highest chances of winning.
D)All of the above.

Question 36

Which of the following outcomes are not mutually exclusive?

A)Flip a fair coin once.Outcomes: head, tail.
B)Flip a fair coin twice.Outcomes: getting at least one tail, getting at least one head.
C)Flip a fair coin twice.Outcomes: getting two tails, getting two heads.
D)All of the above pairs of outcomes are mutually exclusive.

Question 37

For Questions use the following narrative
Narrative: Taxes
Suppose you have a taxing policy where 60% of the population pays $10,000 per year in taxes per person, and the other 40% pays $100 per year in taxes per person.
{Taxes narrative} What is the expected value for the amount of yearly taxes paid by one person in this population?

Question 38

Suppose the outcomes of births within a given family are independent of each other, and a couple has already had four boys.Which of the following best describes the probability that their next baby will be a girl?

A)Approximately 50%
B)Much less than 50%
C)Much greater than 50%
D)Not enough information to tell

Question 39

For Questions use the following narrative:
Narrative: Shaking hands
Suppose the chances of picking up a cold from someone by shaking hands with them is .01 (assuming you don't know whether they have a cold or not), and that each encounter you have is independent of another.
{Shaking hands narrative} Explain, using probability, why your chance of not getting a cold in a given day decreases as the number of handshakes you make increases?

Question 40

For Questions use the following narrative
Narrative: Taxes
Suppose you have a taxing policy where 60% of the population pays $10,000 per year in taxes per person, and the other 40% pays $100 per year in taxes per person.
{Taxes narrative} Explain how the expected value can be misleading if it is interpreted by someone as the amount of taxes they will have to pay in a given year.

Question 41

The __________ represents the average value of any measurement over the long run.

Question 42

The expected value __________ have to be one of the possible outcomes.(Choices: "does" or "does not.")