Statistics Quiz: Using Probability To Make Fair Decisions
20 questions · exam conditions
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Using Probability To Make Fair DecisionsQuestion 1 of 20

Three students—Eli, Fatima, and Gio—volunteer to be class messenger, but only one can be chosen. The class wants a fair method, meaning each student has probability 1/31/3 of being selected. Which method gives each student an equal chance?

Draw a card from a standard deck: hearts = Eli, diamonds = Fatima, clubs or spades = Gio.
Roll a fair 6-sided die: 1–2 = Eli, 3–4 = Fatima, 5–6 = Gio.
Pick whoever raised their hand first.
Flip a fair coin: heads = Eli, tails = choose between Fatima and Gio by rolling a die (1–3 = Fatima, 4–6 = Gio).
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Statistics Quiz

Statistics Quiz: Using Probability To Make Fair Decisions

Practice Using Probability To Make Fair Decisions in Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Using Probability To Make Fair Decisions, giving you a quick way to practice the rules, question types, and explanations that matter most for Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Three students—Eli, Fatima, and Gio—volunteer to be class messenger, but only one can be chosen. The class wants a fair method, meaning each student has probability 1/31/3 of being selected. Which method gives each student an equal chance?

  1. Draw a card from a standard deck: hearts = Eli, diamonds = Fatima, clubs or spades = Gio.
  2. Roll a fair 6-sided die: 1–2 = Eli, 3–4 = Fatima, 5–6 = Gio. (correct answer)
  3. Pick whoever raised their hand first.
  4. Flip a fair coin: heads = Eli, tails = choose between Fatima and Gio by rolling a die (1–3 = Fatima, 4–6 = Gio).
Explanation: This question tests the skill of using probability to make fair decisions. Fairness means that each of the three students has an equal probability of 1/3 of being selected as messenger. In the correct method, choice B, a fair 6-sided die is rolled, with two numbers for each student, so each corresponds to two out of six equally likely outcomes. This assigns equal chances because the die is fair, making each grouping equally probable. One incorrect method, choice A, introduces bias by giving Eli a 1/2 probability from the coin and then splitting the rest unequally with the die. Remember that fairness is about equal chances for each option, not about guaranteeing equal results over multiple trials. To apply this elsewhere, list all possible outcomes and check if each is equally likely.

Question 2

Six students—A, B, C, D, E, and F—are all qualified to lead a lab group, and the teacher wants a fair selection. Fair means each student has probability 1/61/6 of being chosen. Which method is fair?

  1. Roll a fair 6-sided die twice and add: 2=A, 3=B, 4=C, 5=D, 6=E, 7=F (re-roll if you get 8–12).
  2. Choose the student who sits closest to the door so it's easy for them to leave the room.
  3. Flip a coin: heads = pick from A, B, C (teacher chooses), tails = pick from D, E, F (teacher chooses).
  4. Use a spinner with 6 equal sections labeled A–F, and spin once. (correct answer)
Explanation: This question tests the skill of using probability to make fair decisions. Fairness means that each of the six students has an equal probability of 1/6 of being chosen to lead. In the correct method, choice A, a spinner with six equal sections labeled A through F is spun once, creating six equally likely landing spots. This assigns equal chances because the sections are equal, making each outcome equally probable. One incorrect method, choice B, introduces bias by using dice sums that are not equally likely and re-rolling higher sums, leading to unequal probabilities like 1/21 for A and 6/21 for F. Remember that fairness is about equal chances for each option, not about guaranteeing equal results over multiple trials. To apply this elsewhere, list all possible outcomes and check if each is equally likely.

Question 3

Five friends—Kai, Luz, Mira, Omar, and Priya—need a fair way to decide who gets the first turn in a board game. Fair means each friend has probability 1/51/5 of being selected. Which method is fair?

  1. Let the oldest player go first because that seems respectful.
  2. Roll a fair 6-sided die: 1 = Kai, 2 = Luz, 3 = Mira, 4 = Omar, 5 or 6 = Priya.
  3. Spin a spinner with 5 equal sections, but if it lands on Priya you re-spin because Priya went first last time.
  4. Put 5 identical cards labeled with the names into a bag, shake, and draw 1 card. (correct answer)
Explanation: This question tests the skill of using probability to make fair decisions. Fairness means that each of the five friends has an equal probability of 1/5 of getting the first turn. In the correct method, choice B, five identical cards with the names are put in a bag, shaken, and one drawn, so there are five equally likely cards to select. This assigns equal chances because each card is identical and the shaking ensures random selection. One incorrect method, choice A, introduces bias by assigning two die outcomes to Priya and one to each other, leading to probabilities of 1/3 for Priya and 1/6 for others. Remember that fairness is about equal chances for each option, not about guaranteeing equal results over multiple trials. To apply this elsewhere, list all possible outcomes and check if each is equally likely.

Question 4

Four songs—W, X, Y, and Z—are finalists for the school dance playlist, and the DJ needs a fair way to pick the first song. Fair means each song has probability 1/41/4 of being chosen. Which procedure is fair?

  1. Pick the song that most students say they "like," because that feels fair to the group.
  2. Flip two fair coins: if you get exactly one head choose W, otherwise choose among X, Y, Z by the DJ's preference.
  3. Spin a spinner labeled W, X, Y, Z where the W section is half the circle and the other three sections split the remaining half equally.
  4. Use a random number generator that outputs 1, 2, 3, or 4 with equal probability; assign 1=W, 2=X, 3=Y, 4=Z. (correct answer)
Explanation: This question tests the skill of using probability to make fair decisions. Fairness means that each of the four songs has an equal probability of 1/4 of being chosen first. In the correct method, choice A, a random number generator outputs 1 through 4 with equal probability, each assigned to one song, creating four equally likely outcomes. This assigns equal chances because the generator ensures each number is equally probable. One incorrect method, choice B, introduces bias by making the W section half the spinner, resulting in 1/2 probability for W and 1/6 for each other. Remember that fairness is about equal chances for each option, not about guaranteeing equal results over multiple trials. To apply this elsewhere, list all possible outcomes and check if each is equally likely.

Question 5

Four students—Ava, Ben, Cara, and Diego—need a fair way to choose who presents first. Fair means each student has an equal probability (1/41/4) of being selected, even if the same person could be chosen on different days. Which procedure is fair?

  1. Have the teacher pick the student whose last name comes first alphabetically.
  2. Write each name on an identical slip of paper, mix the slips thoroughly in a cup, and draw 1 slip. (correct answer)
  3. Put 3 blue marbles and 1 red marble in a bag; each student is assigned a color, and the student with the drawn color presents first.
  4. Flip a coin: heads means Ava or Ben presents (teacher chooses which), tails means Cara or Diego presents (teacher chooses which).
Explanation: This question tests the skill of using probability to make fair decisions. Fairness means that each of the four students has an equal probability of 1/4 of being selected to present first. In the correct method, choice B, each student's name is written on an identical slip, mixed in a cup, and one is drawn, so there are four equally likely outcomes corresponding to the four names. This assigns equal chances because each slip has the same probability of being drawn due to thorough mixing. One incorrect method, choice A, introduces bias by using marbles of different colors with unequal numbers, likely leading to unequal selection probabilities depending on color assignments. Remember that fairness is about equal chances for each option, not about guaranteeing equal results over multiple trials. To apply this elsewhere, list all possible outcomes and check if each is equally likely.

Question 6

Three movie options—Comedy, Action, and Documentary—are being considered for a class reward day. The class wants a fair method to select one, meaning each option should have probability 1/31/3 of being chosen. Which procedure is fair?

  1. Vote, and if there is no majority, the teacher chooses the one they prefer.
  2. Roll a fair 6-sided die: 1–2 = Comedy, 3–4 = Action, 5–6 = Documentary. (correct answer)
  3. Spin a spinner with 3 sections labeled Comedy, Action, Documentary, but the Documentary section is slightly smaller.
  4. Flip a coin: heads = Comedy, tails = re-flip until you get heads, then choose between Action and Documentary by a class discussion.
Explanation: This question tests the skill of using probability to make fair decisions. Fairness means that each of the three movie options has an equal probability of 1/3 of being selected. In the correct method, choice B, a fair 6-sided die is rolled, with two numbers for each option, so each corresponds to two out of six equally likely outcomes. This assigns equal chances because the die is fair, ensuring equal probability for each grouping. One incorrect method, choice C, introduces bias by making the Documentary section smaller on the spinner, leading to a lower probability for it. Remember that fairness is about equal chances for each option, not about guaranteeing equal results over multiple trials. To apply this elsewhere, list all possible outcomes and check if each is equally likely.

Question 7

Three teams (Red, Blue, Green) are tied and must be assigned to present in the first time slot. The method is fair only if each team has probability 13\tfrac13 of being chosen for the first slot (fairness is about equal chances, not equal outcomes). Which procedure is fair?

  1. Roll a fair six-sided die: 1–2 = Red, 3–4 = Blue, 5–6 = Green. (correct answer)
  2. Choose the team whose name comes first alphabetically.
  3. Flip a coin: heads = Red; tails = spin a spinner that has equal halves Blue and Green.
  4. Roll a fair six-sided die: 1–3 = Red, 4–5 = Blue, 6 = Green.
Explanation: The skill here is using probability to make fair decisions among three teams: Red, Blue, and Green. Fairness means each team has an equal probability of 1/3 of being chosen for the first slot. In the correct method, a fair six-sided die is rolled, assigning 1–2 to Red, 3–4 to Blue, and 5–6 to Green. This assigns equal chances because each pair of numbers has the same two outcomes out of six, making each team's probability 2/6 or 1/3. One incorrect method is rolling a die with 1–3 for Red, 4–5 for Blue, and 6 for Green, which introduces bias by giving Red a higher probability of 3/6. Remember, fairness is about equal chances for each selection, not about equal outcomes over multiple trials. To apply this elsewhere, list all possible outcomes and check if each option is equally likely.

Question 8

Two movies (Movie 1 and Movie 2) are tied in a class vote, and the class wants a fair tie-breaker. Fair means each movie has probability 12\tfrac12 of being selected, regardless of what happens on any one day. Which procedure is fair?

  1. Choose Movie 1 because it got the same number of votes as Movie 2, so either choice is equally fair.
  2. Roll a die: 1–4 = Movie 1, 5–6 = Movie 2.
  3. Flip a fair coin once: heads = Movie 1, tails = Movie 2. (correct answer)
  4. Spin a spinner that looks split into two parts, but one part is slightly larger; choose the movie the spinner lands on.
Explanation: The skill here is using probability to make fair decisions between two movies: Movie 1 and Movie 2. Fairness means each movie has an equal probability of 1/2 of being selected. In the correct method, a fair coin is flipped once, with heads for Movie 1 and tails for Movie 2. This assigns equal chances because the coin has two sides, each with a probability of 1/2. One incorrect method is rolling a die with 1–4 for Movie 1 and 5–6 for Movie 2, which introduces bias by giving Movie 1 a higher probability of 4/6. Remember, fairness is about equal chances for each selection, not about equal outcomes over multiple trials. To apply this elsewhere, list all possible outcomes and check if each option is equally likely.

Question 9

Three students—Hana, Isaac, and Jorge—need a fair way to decide who cleans up lab equipment today. The method is fair if each has probability 13\tfrac13 of being chosen. Which procedure is fair?

  1. Draw one card from a well-shuffled set of 3 cards labeled Hana, Isaac, Jorge (one card each). (correct answer)
  2. Roll a die: 1–2 = Hana, 3–4 = Isaac, 5 = Jorge, and re-roll if 6.
  3. Flip a coin: heads = Hana, tails = Isaac; if it lands on its edge, Jorge.
  4. Choose the student who cleaned up least recently, so the results stay balanced.
Explanation: The skill here is using probability to make fair decisions among three students: Hana, Isaac, and Jorge. Fairness means each has an equal probability of 1/3 of being chosen to clean up. In the correct method, one card labeled with each name is shuffled, and one is drawn. This assigns equal chances because each card is equally likely to be drawn, at 1/3 each. One incorrect method is rolling a die with 1–2 for Hana, 3–4 for Isaac, 5 for Jorge, and re-roll on 6, which introduces bias by giving Hana and Isaac higher probabilities of about 2/5 each and Jorge 1/5. Remember, fairness is about equal chances for each selection, not about equal outcomes over multiple trials. To apply this elsewhere, list all possible outcomes and check if each option is equally likely.

Question 10

Four students—Ava, Ben, Chloe, and Diego—need a fair way to choose who presents first. A procedure is fair if each student has the same probability (each should have a 14\tfrac14 chance), even if the outcome isn't equal over time. Which procedure is fair?

  1. Go in alphabetical order by first name (Ava, then Ben, then Chloe, then Diego).
  2. Draw one name from a bag that contains 2 slips for Ava, 1 for Ben, 1 for Chloe, and 1 for Diego.
  3. Spin a spinner with four equal-sized sections labeled Ava, Ben, Chloe, Diego once; whoever it lands on presents first. (correct answer)
  4. Flip a coin: if heads, Ava presents; if tails, randomly choose among Ben, Chloe, and Diego by rolling a die and re-rolling 5 or 6.
Explanation: The skill here is using probability to make fair decisions among four students: Ava, Ben, Chloe, and Diego. Fairness means each student has an equal probability of 1/4 of being chosen to present first. In the correct method, a spinner with four equal-sized sections labeled with each name is spun once, and the landing section determines the presenter. This assigns equal chances because each section has the same size, so the probability of landing on any one is identical at 1/4. One incorrect method is drawing from a bag with uneven slips, like two for Ava and one each for others, which introduces bias by giving Ava a higher probability of 2/5. Remember, fairness is about equal chances for each selection, not about equal outcomes over multiple trials. To apply this elsewhere, list all possible outcomes and check if each option is equally likely.

Question 11

Five volunteers—Jules, Kai, Lina, Mo, and Noor—will be chosen to lead warm-ups today. The coach wants a fair selection, meaning each person has probability 1/51/5 of being chosen (even though the outcome will pick only one person). Which procedure is fair?

  1. Pick the person who volunteered first.
  2. Use a random number generator to pick an integer from 1 to 10; 1–2=Jules, 3–4=Kai, 5–6=Lina, 7–8=Mo, 9–10=Noor. (correct answer)
  3. Draw a marble from a bag with 2 red, 2 blue, 2 green, 2 yellow, and 2 purple marbles; assign Jules=red, Kai=blue, Lina=green, Mo=yellow, Noor=purple, and redraw if you pick purple.
  4. Roll a fair 6-sided die: 1=Jules, 2=Kai, 3=Lina, 4=Mo, 5=Noor, and re-roll on 6 but keep re-rolling until Jules has been selected at least once today.
Explanation: This question involves using probability to make fair decisions among five volunteers, where fairness means each person has exactly 1/5 probability of selection. To verify fairness, we examine the probability distribution for each method. Option A uses a random number generator with 10 outcomes, assigning 2 to each person (Jules: 1-2, Kai: 3-4, Lina: 5-6, Mo: 7-8, Noor: 9-10), giving each person 2/10 = 1/5 probability. Option B uses volunteering order, which isn't random and creates timing bias. Option C's rule to keep re-rolling until Jules is selected changes the probabilities, creating selection bias that favors Jules. Option D assigns equal marble colors but redraws if purple (Noor) is picked, reducing Noor's chances below 1/5 and creating color bias. Remember that fairness concerns the chances in the selection process, not the actual outcome.

Question 12

Two books—Book X and Book Y—are equally popular, and the librarian needs a fair tie-breaker to decide which one gets displayed first. Fair means each book has probability 1/21/2 of being chosen. Which procedure is fair?

  1. Flip a fair coin: heads=Book X, tails=Book Y. (correct answer)
  2. Ask three staff members to vote; whichever book gets more votes wins.
  3. Roll a fair 6-sided die: 1–4=Book X and 5–6=Book Y.
  4. Display the book with the earlier publication year.
Explanation: This problem tests using probability to make fair decisions between two books, where fairness means each book has exactly 1/2 probability. To determine fairness, we calculate probabilities for each option. Option B uses a fair coin flip, giving heads=Book X and tails=Book Y, so each book has exactly 1/2 probability - this is fair. Option A uses publication year, which isn't random and creates chronological bias favoring the older book. Option C assigns 4 die outcomes to Book X (1-4) and 2 to Book Y (5-6), giving Book X a 4/6 = 2/3 chance and creating numerical bias. Option D uses voting, which depends on staff preferences and creates preference bias. Fairness requires equal chances for both books through a random process, not considerations of age or popularity.

Question 13

A club has four project ideas—Garden, Mural, Recycling, and Tutoring—and must pick one to start first. They want a fair random choice, meaning each idea has probability 1/41/4 of being selected. Which method gives each idea an equal chance?

  1. Roll two fair dice and add: 2–4=Garden, 5–7=Mural, 8–10=Recycling, 11–12=Tutoring.
  2. Flip a coin: heads=Garden, tails=Recycling; if someone complains, switch to Mural.
  3. Pick the idea that appears first in the club's written list.
  4. Use a random number generator to pick 1, 2, 3, or 4 with equal probability; 1=Garden, 2=Mural, 3=Recycling, 4=Tutoring. (correct answer)
Explanation: This question tests using probability to make fair decisions among four project ideas, where fairness means each idea has exactly 1/4 probability. To determine fairness, we examine the probability distribution for each method. Option C uses a random number generator to pick 1, 2, 3, or 4 with equal probability, assigning one number to each project, so each has exactly 1/4 chance - this is fair. Option A uses list order, which isn't random and creates positional bias. Option B uses dice sums with unequal ranges (Garden: 3 outcomes, Mural: 3 outcomes, Recycling: 3 outcomes, Tutoring: 2 outcomes out of 11 possible sums), creating sum bias. Option D only considers two projects initially and adds subjective switching, creating selection bias. Fairness requires equal chances for all four ideas through a truly random process.

Question 14

Three snack options—apples, chips, and cookies—are available, and the group wants to pick one in a fair way so each snack has probability 13\tfrac{1}{3} of being chosen. Which method gives each snack an equal chance?

  1. Roll a die: 1–2=apples, 3–4=chips, 5–6=cookies. (correct answer)
  2. Roll a die: 1–3=apples, 4–5=chips, 6=cookies.
  3. Choose cookies because they are the most popular, so everyone feels included.
  4. Pick the snack that starts with the earliest letter in the alphabet.
Explanation: The skill here is using probability to make fair decisions. Fairness means that each snack has an equal probability of being selected, specifically 1/3 for apples, chips, and cookies. In the correct method, a die is rolled with 1–2 for apples, 3–4 for chips, and 5–6 for cookies. This assigns equal chances because each pair of numbers is equally likely, giving each snack two out of six outcomes for a probability of 1/3. One incorrect method is assigning 1–3 to apples, 4–5 to chips, and 6 to cookies, introducing bias by giving apples 1/2, chips 1/3, and cookies 1/6. Remember, fairness is about equal chances in each selection, not about the same snack being chosen equally often over multiple selections. To apply this strategy elsewhere, list all possible outcomes and check if each option is equally likely.

Question 15

A class must choose one of four projects—A, B, C, or D—to display in the hallway. The teacher wants a fair random choice, meaning each project has probability 14\tfrac{1}{4} of being selected. Which procedure is fair?

  1. Put four identical slips labeled A, B, C, and D into a bag, mix well, and draw one slip. (correct answer)
  2. Roll a die: 1=A, 2=B, 3=C, 4=D, and if 5 or 6 occurs, pick the teacher's favorite.
  3. Choose the project that has the neatest handwriting so the display looks best.
  4. Spin a spinner labeled A, B, C, D where the A section covers half the circle and the other three share the remaining half equally.
Explanation: The skill here is using probability to make fair decisions. Fairness means that each project has an equal probability of being selected, specifically 1/4 for A, B, C, and D. In the correct method, four identical slips labeled A, B, C, and D are put in a bag, mixed, and one is drawn. This assigns equal chances because each slip is identical and equally likely to be drawn, so each project has a probability of 1/4. One incorrect method is spinning a spinner where A covers half the circle and the others share the rest, introducing bias by giving A a probability of 1/2 and the others 1/6 each. Remember, fairness is about equal chances in each selection, not about the same project being chosen equally often over multiple selections. To apply this strategy elsewhere, list all possible outcomes and check if each option is equally likely.

Question 16

Six students—Fay, Gus, Hana, Ivan, June, and Leo—need a fair way to decide who cleans the whiteboard. Fair means each student has probability 16\tfrac{1}{6} of being chosen. Which method is fair?

  1. Roll a standard die once and assign 1=Fay, 2=Gus, 3=Hana, 4=Ivan, 5=June, 6=Leo. (correct answer)
  2. Roll a standard die: 1–2=Fay, 3=Gus, 4=Hana, 5=Ivan, 6=June, and if Leo is rolled, reroll.
  3. Pick the student who is closest to the board to save time.
  4. Have the students draw straws, but make Fay's straw slightly shorter so it is easier to tell apart.
Explanation: The skill here is using probability to make fair decisions. Fairness means that each student has an equal probability of being selected, specifically 1/6 for Fay, Gus, Hana, Ivan, June, and Leo. In the correct method, a standard die is rolled once, assigning each number 1–6 to one student. This assigns equal chances because the die has six equally likely faces, so each student has a probability of 1/6. One incorrect method is assigning 1–2 to Fay, single numbers to others, and rerolling on an unassigned 'Leo' roll, introducing bias by giving Fay a higher probability than the others. Remember, fairness is about equal chances in each selection, not about the same student being chosen equally often over multiple selections. To apply this strategy elsewhere, list all possible outcomes and check if each option is equally likely.

Question 17

Two candidates, Sam and Taylor, are tied for class representative. The class wants a fair decision rule, meaning each candidate has probability 12\tfrac{1}{2} of winning. Which procedure is fair?

  1. Pick the candidate whose last name comes first alphabetically.
  2. Roll a standard die: 1–3 means Sam wins and 4–6 means Taylor wins. (correct answer)
  3. Flip a coin, but if it lands on heads, flip again to confirm before choosing Sam.
  4. Let Sam win today and promise Taylor will win the next tie, so the outcomes are balanced.
Explanation: The skill here is using probability to make fair decisions. Fairness means that each candidate has an equal probability of being selected, specifically 1/2 for Sam and Taylor. In the correct method, a standard die is rolled, with 1–3 for Sam and 4–6 for Taylor. This assigns equal chances because there are three numbers for each, out of six equally likely outcomes, so each has a probability of 1/2. One incorrect method is flipping a coin but confirming heads again before choosing Sam, introducing bias by potentially altering the probability away from 1/2 for each. Remember, fairness is about equal chances in each selection, not about the same candidate being chosen equally often over multiple selections. To apply this strategy elsewhere, list all possible outcomes and check if each option is equally likely.

Question 18

Three students—Ava, Ben, and Carlos—must be chosen to present first in class. The teacher wants the choice to be fair, meaning each student has an equal probability of being selected (not that the same student is chosen equally often over time). Which procedure is fair?

  1. Spin a spinner with three equal-sized sections labeled Ava, Ben, and Carlos once; whoever it lands on presents first. (correct answer)
  2. Spin a spinner labeled Ava, Ben, and Carlos, but Ava's section is twice as large as each of the others.
  3. Have the students line up alphabetically by first name; the first in line presents first.
  4. Flip a coin: heads means Ava presents first; tails means choose between Ben and Carlos by asking who volunteers.
Explanation: The skill here is using probability to make fair decisions. Fairness means that each student has an equal probability of being selected, specifically 1/3 for Ava, Ben, and Carlos. In the correct method, a spinner with three equal-sized sections labeled for each student is spun once, and the landing section determines the presenter. This assigns equal chances because each section has the same area, so the probability of landing on any one is 1/3. One incorrect method is spinning a spinner where Ava's section is twice as large, introducing bias by giving Ava a higher probability of 1/2 while Ben and Carlos each have 1/4. Remember, fairness is about equal chances in each selection, not about the same student being chosen equally often over multiple selections. To apply this strategy elsewhere, list all possible outcomes and check if each option is equally likely.

Question 19

Two classes, Class 1 and Class 2, are tied for which class gets to use the gym first. They want a fair random method. Fair means each class has probability 12\tfrac12 of being chosen. Which procedure is fair?

  1. Roll a standard die: 1–3 = Class 1, 4–6 = Class 2. (correct answer)
  2. Roll a standard die: 1–4 = Class 1, 5–6 = Class 2.
  3. Let Class 1 go first this time because Class 2 went first last time, so the outcomes will be equal overall.
  4. Flip a coin, but if it lands on tails, flip again and use the second result (heads = Class 1, tails = Class 2).
Explanation: This problem tests using probability to make fair decisions. Fairness means each class has an equal probability of being chosen—in this case, 1/2 for each of the two classes. Rolling a die with 1-3 for Class 1 and 4-6 for Class 2 (choice A) gives each class exactly 3/6 = 1/2 probability since each die outcome is equally likely. This method ensures equal chances because each class is assigned the same number of equally likely die outcomes. The 1-4 vs 5-6 assignment (choice B) creates bias by giving Class 1 probability 4/6 = 2/3 and Class 2 only 1/3. Taking turns (choice C) isn't random, while the double-flip method (choice D) is unnecessarily complex. Remember that fairness is about equal probability of selection each time, not alternating or evening out results. To check fairness with two options, verify each has probability 1/2.

Question 20

Three teams—Red, Blue, and Green—are tied and need a fair tie-breaker to decide who gets the first choice of practice time. Fair means each team has probability 13\tfrac13 of being selected. Which method gives each team an equal chance?

  1. Roll a standard die: 1–2 = Red, 3–4 = Blue, 5–6 = Green. (correct answer)
  2. Roll a standard die: 1–3 = Red, 4–5 = Blue, 6 = Green.
  3. Pick the team that has waited the longest in previous weeks so the results "even out."
  4. Flip a coin: heads = Red, tails = Blue; if it lands on tails, flip again to decide between Blue and Green.
Explanation: This question involves using probability to make fair decisions. Fairness means each team has an equal probability of being selected—here, 1/3 for each of the three teams. Rolling a die with assignments 1-2=Red, 3-4=Blue, 5-6=Green (choice A) gives each team exactly 2/6 = 1/3 probability since each die outcome is equally likely. This method ensures equal chances because each team is assigned the same number of equally likely die outcomes. The incorrect method in choice B assigns Red half the outcomes (3/6), creating bias toward Red, while choice D's coin flip method gives Red 1/2 probability but Blue and Green each get 1/4. Fairness concerns the probability of selection, not making results "even out" over time. To verify fairness, list all possible outcomes and check that each option appears with equal frequency.