Question 1

Football is a very popular sport in the United States.The National Football League (NFL)is composed of 32 teams.Every year there is a football draft where players are selected for each of the teams.Football players in the United States can make millions of dollars and must be very good at the sport to make a team.Would being selected for an NFL team be considered a random phenomenon?
A)Yes,everyone is probably very good at football and would qualify for a team.
B)Yes,there are a large number of teams,which makes this a random phenomenon.
C)No,there are probably certain skill sets necessary for football that owners of teams are looking for in players.
D)No,if you really want to play football,you can probably join an NFL team.

Accepted Answer

Being selected for an NFL team is not a random phenomenon as there are specific skill sets that teams are looking for in players. It takes years of training and dedication to have the necessary skills to make a team, and not everyone who wants to play can simply join an NFL team. The draft may seem random, but teams are strategically selecting players based on their abilities and the team's needs.

Question 2

Which of the following is/are appropriate statements about randomness and/or probability?
A)A phenomenon is called random if individual outcomes are uncertain,but in a large number of repetitions,there is a regular distribution of outcomes.
B)The word random in statistics is a description of a kind of order that emerges in the long run.
C)Probability describes only what happens in the long run.
D)In a small or moderate number of repetitions,the observed proportion of an outcome can be far from the probability of the outcome.
E)All of the above are appropriate statements.

Accepted Answer

Each statement accurately reflects concepts related to randomness and probability. A) and B) describe the nature of randomness as having a predictable distribution over many trials, despite individual outcomes being uncertain. C) highlights that probability is concerned with long-term behavior, and D) acknowledges that short-term outcomes can deviate significantly from long-term probabilities.

Question 3

The probability of an event is defined as the proportion of times an event occurs occasionally.

Accepted Answer

The probability of an event is defined as the number of favorable outcomes divided by the total number of possible outcomes, not just the proportion of times the event occurs.

Question 4

In a certain game you have to guess the number that your opponent writes down on a sheet of paper.You get five guesses.After each guess,your opponent has to tell you if your number is too high or too low.Each guess is considered ________ the last guess.
A)independent of
B)dependent on

Accepted Answer

Each guess is not independent, as the feedback from the opponent on whether the previous guess was too high or too low affects the next guess. Therefore, option B, "dependent on," is the correct choice.

Question 5

There is a popular pizza restaurant on campus that many students frequent.One evening,your college sorority goes to the restaurant to celebrate the end of the semester.At the end of the evening,you are not feeling well and several of your sorority sisters do not feel well either.You suspect food poisoning from the restaurant is the culprit.Suppose your roommate is also sick and ate at the same restaurant on the same day as you.Would your roommate being sick be an example of a random phenomenon?
A)Yes,because more than one person is sick it must be a random phenomenon.
B)Yes,because you went to a random restaurant on campus.
C)No,since your roommate ate at the same restaurant,it is probably not a random phenomenon that she is sick.
D)No,everyone at the restaurant would have to be sick for it to be a random phenomenon.

Accepted Answer

The fact that your roommate is also sick and ate at the same restaurant on the same day indicates that it may not be a random phenomenon. It suggests that the source of the illness may be linked to the restaurant, rather than purely by chance.

Question 6

You decide you want to start a garden and grow tomatoes and cucumbers.After a month the yield of the tomatoes and cucumbers is quite low.You dig up the plants and start again in the same plot.However,the next month the yield of the tomatoes and cucumbers is still quite low.The yield of the tomatoes and cucumbers from month 1 to month 2 is considered __________.
A)independent
B)dependent

Accepted Answer

The yield of the tomatoes and cucumbers in the second month is dependent on the yield in the first month, as replanting in the same plot does not guarantee a significantly different outcome.

Question 7

The statistics of a particular basketball player state that he makes 4 out of 5 free-throw attempts.During a season,the basketball player makes an average of about 200 free-throw attempts.On how many of these throws would you estimate he will actually score a point?
A)80
B)100
C)120
D)160

Accepted Answer

Given the player's success rate is 4 out of 5 attempts, this translates to an 80% success rate. For 200 attempts, 80% of 200 is 160.

Question 8

Checkers is a board game played between two players,who alternate moves.The board is square,with sixty-four smaller squares,arranged in an eight-by-eight grid.The smaller squares are alternately light and dark colored,in the famous "checker-board" pattern.Each player starts with 12 pieces.You win by capturing all your opponent's pieces.This is accomplished by jumping over a piece,diagonally,to the adjacent vacant square beyond the piece.If each player's goal is to win the game,then each move is considered _________.
A)independent
B)dependent

Accepted Answer

Each move in Checkers is dependent on the position of the pieces on the board, the moves made by the opponent, and the rules of the game, making strategy and reaction to the opponent's moves crucial.

Question 9

A phenomenon is observed many,many times under identical conditions.The proportion of times a particular event A occurs is recorded.What does this proportion represent?
A)The probability of the event A
B)The distribution of the event A
C)The correlation of the event A
D)The variance of the event A

Accepted Answer

The proportion of times a particular event A occurs under identical conditions is a measure of its frequency. As the phenomenon is observed many times, the proportion of occurrences of event A becomes a reliable estimate of the probability of event A occurring. Therefore, the proportion represents the probability of the event A.

Question 10

For an event to be random (e.g. ,flipping a coin five times),a cluster of the same observations (e.g. ,all heads)would not be possible.

Accepted Answer

Random events can result in clusters of the same observations due to the nature of probability, meaning it is possible to get all heads when flipping a coin five times.

Question 11

If you have a fair die,what is the expected proportion of rolls that result in a number greater than 4?
A)1/6
B)1/3
C)2/3
D)1/2

Accepted Answer

The numbers greater than 4 on a fair die are 5 and 6, which are 2 out of the 6 possible outcomes. Therefore, the expected proportion is 2/6, which simplifies to 1/3.

Question 12

In a certain game you have to guess the number that your opponent writes down on a sheet of paper.You get five guesses and no additional information is given after each guess unless it is the correct number.Each guess is considered _______ the last guess.
A)independent of
B)dependent on

Accepted Answer

Each guess is independent of the last guess, meaning that the probability of guessing the correct number does not change based on the previous guess. The previous guess does not provide any additional information to help or hinder the next guess.

Question 13

The game of craps starts with a "come-out" roll,in which the shooter rolls a pair of dice.If the total of the "spots" on the up-faces is 2,3,or 12,the shooter loses immediately.What is the proportion of come-out rolls that ends up in an immediate loss for the shooter? A)1/9 B)¹/₄ C)8/9 D)1/12

Accepted Answer

The shooter loses immediately on a total of 2, 3, or 12. There are 36 possible outcomes when rolling two dice (6 faces on the first die times 6 faces on the second die). The combinations that result in a total of 2, 3, or 12 are: one combination for 2 (1+1), two combinations for 3 (1+2, 2+1), and one combination for 12 (6+6). This gives a total of 4 losing combinations out of 36, which simplifies to 1/9.

Question 14

The statistics of a particular basketball player state that he makes 4 out of 5 free-throw attempts.The basketball player is just about to attempt a free throw.What do you estimate the probability that the player makes this next free throw to be?
A)0.16
B)50-50-either he makes it or he doesn't.
C)0.80
D)1.2

Accepted Answer

The player has a 4/5 or 0.80 probability of making a free throw based on the given statistics.

Question 15

A penny is tossed.We observe whether it lands heads up or tails up.Suppose the penny is a fair coin;that is,the probability of heads is one-half and the probability of tails is one-half.What does this mean?
A)Every occurrence of a head must be balanced by a tail in one of the next two or three tosses.
B)If the coin is tossed many,many times,the proportion of tosses that land heads will be approximately one-half,and this proportion will tend to get closer and closer to one-half as the number of tosses increases.
C)Regardless of the number of flips,one-half will be heads and one-half tails.
D)All of the above

Accepted Answer

This means that over the long run, we can expect approximately half of the coin tosses to land heads and half to land tails. It does not mean that every occurrence of a head must be balanced by a tail in the next few tosses or that every flip will always result in exactly half heads and half tails.

Question 16

Which of the following best describes what is meant by the probability of the outcome of a random phenomenon?
A)The count of the short-term occurrences of the outcomes
B)The count of the outcomes of the phenomenon that occur in a haphazard manner
C)The proportion of outcomes that can be expected to be observed during a short experiment
D)The proportion of times the outcome would occur in a very long series of repetitions
E)The proportion of repetitions that result in the outcome when an experiment is conducted about the random phenomenon

Accepted Answer

The probability of the outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. This is because the probability is based on the long-term frequency of the occurrence, not just the short-term occurrences or haphazard occurrences.

Question 17

There are five upper-level mathematics courses that are required for math majors at a certain university.You can only pass the last course after you have passed all of the other four.Passing the last course is considered a(n)________ event.
A)independent
B)dependent

Accepted Answer

Passing the last course depends on passing all of the other four courses, so it is a dependent event.

Question 18

You are taking a basic statistics class.On the first exam you score poorly and receive only 10 out of 100 points.You decide to get a tutor for the next exam and study quite a bit more.You end up scoring much better and receive 80 out of 100 points.The scores on your first and second exam are considered _________.
A)independent
B)dependent

Accepted Answer

The scores on the first and second exam are considered independent because the outcome of one exam does not affect the outcome of the other exam.

Question 19

To win the Maryland Powerball^® Lottery,you must select your own six numbers to play.Most people use numbers that are special to them,like anniversary dates or their children's birth dates.On the day of the lottery,the lottery officials randomly select five white balls,each with a number somewhere between 1 and 59,and one red Powerball^® with a number somewhere between 1 and 35.You win a prize by matching some or all of the numbers drawn.Would winning the lottery be an example of a random phenomenon? A)Yes,because the numbers drawn are random B)Yes,because you are selecting your own six numbers to play C)No,because the numbers you select to play are not random if you choose your anniversary date or your children's birth dates D)No,because the numbers drawn are random

Accepted Answer

Winning the lottery involves the random selection of numbers by lottery officials. While players may select their own numbers, the outcome is ultimately determined by random chance.

Question 20

Which of the following are independent trials?
A)The results of a coin flip after rolling two dice
B)The first character of a license plate and the color of the car
C)The total of two dice after recording the value of the first die rolled
D)A and B
E)All of the above

Accepted Answer

Independent trials are those where the outcome of one trial does not affect the outcome of another. A) The result of a coin flip is independent of rolling two dice, and B) The first character of a license plate is independent of the color of the car. However, C) is not independent because knowing the value of the first die can affect the total outcome of the two dice.

Quiz 4: Probability: the Study of Randomness

Which of the following is/are appropriate statements about randomness and/or probability?

The probability of an event is defined as the proportion of times an event occurs occasionally.

In a certain game you have to guess the number that your opponent writes down on a sheet of paper.You get five guesses.After each guess,your opponent has to tell you if your number is too high or too low.Each guess is considered ________ the last guess.

The statistics of a particular basketball player state that he makes 4 out of 5 free-throw attempts.During a season,the basketball player makes an average of about 200 free-throw attempts.On how many of these throws would you estimate he will actually score a point?

A phenomenon is observed many,many times under identical conditions.The proportion of times a particular event A occurs is recorded.What does this proportion represent?

For an event to be random (e.g. ,flipping a coin five times),a cluster of the same observations (e.g. ,all heads)would not be possible.

If you have a fair die,what is the expected proportion of rolls that result in a number greater than 4?

In a certain game you have to guess the number that your opponent writes down on a sheet of paper.You get five guesses and no additional information is given after each guess unless it is the correct number.Each guess is considered _______ the last guess.

The game of craps starts with a "come-out" roll,in which the shooter rolls a pair of dice.If the total of the "spots" on the up-faces is 2,3,or 12,the shooter loses immediately.What is the proportion of come-out rolls that ends up in an immediate loss for the shooter?

The statistics of a particular basketball player state that he makes 4 out of 5 free-throw attempts.The basketball player is just about to attempt a free throw.What do you estimate the probability that the player makes this next free throw to be?

A penny is tossed.We observe whether it lands heads up or tails up.Suppose the penny is a fair coin;that is,the probability of heads is one-half and the probability of tails is one-half.What does this mean?

Which of the following best describes what is meant by the probability of the outcome of a random phenomenon?

There are five upper-level mathematics courses that are required for math majors at a certain university.You can only pass the last course after you have passed all of the other four.Passing the last course is considered a(n) ________ event.

Which of the following are independent trials?