Statistics Quiz: Using Probability To Make Fair Decisions

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.

Question 1

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

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

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

1. Pick the student whose first name has fewer letters.
2. Jada gets the seat today, and Noor gets it tomorrow, so the outcomes are equal over two days.
3. Flip a fair coin once: heads = Jada gets the seat, tails = Noor gets the seat. (correct answer)
4. Roll a fair 6-sided die: 1–4 = Jada, 5–6 = Noor.

Explanation: This question tests the skill of using probability to make fair decisions. Fairness means that each of the two students has an equal probability of 1/2 of getting the window seat. In the correct method, choice B, a fair coin is flipped once, with heads for Jada and tails for Noor, creating two equally likely outcomes. This assigns equal chances because the coin is fair, giving each side the same probability. One incorrect method, choice D, introduces bias by assigning four die outcomes to Jada and two to Noor, resulting in probabilities of 2/3 and 1/3. 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

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 6

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 7

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 8

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 9

1. Roll a fair die: if 1–6, choose that player number; if 7 is needed, re-roll until a 6 appears and then choose player 7.
2. Choose the tallest player to avoid arguments.
3. Use a random number generator from 1 to 8; assign 1–7 to players 1–7 and redraw if it shows 8, but redraw twice if it shows 1.
4. Put 7 identical slips (one per player) into a bag, mix thoroughly, and draw one slip. (correct answer)

Explanation: The skill here is using probability to make fair decisions among seven players: 1 through 7. Fairness means each has an equal probability of 1/7 of being chosen as captain. In the correct method, seven identical slips, one per player, are put in a bag, mixed, and one is drawn. This assigns equal chances because each slip is equally likely to be drawn, at 1/7 each. One incorrect method is using a random number from 1 to 8, assigning 1–7 to players and redrawing on 8 but redrawing twice on 1, which introduces bias by altering the probability for player 1. 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

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 11

1. Flip a fair coin once: heads = Jada, tails = Malik. (correct answer)
2. Jada goes first today because Malik went first yesterday.
3. Flip a coin until it lands on heads; if the first heads is on an odd-numbered flip, Jada gets the seat, otherwise Malik.
4. Roll a fair die: 1–4 = Jada, 5–6 = Malik.

Explanation: The skill here is using probability to make fair decisions between two classmates: Jada and Malik. Fairness means each has an equal probability of 1/2 of getting the seat. In the correct method, a fair coin is flipped once, with heads for Jada and tails for Malik. 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 Jada and 5–6 for Malik, which introduces bias by giving Jada 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 12

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 13

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 14

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 15

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 16

1. Roll a fair 8-sided die labeled 1–8 and match the number to P1–P8. (correct answer)
2. Roll a fair 6-sided die: 1=P1, 2=P2, 3=P3, 4=P4, 5=P5, 6=P6; if you roll 6, then pick randomly between P7 and P8.
3. Let the youngest player serve first to keep it fair to the younger players.
4. Flip a fair coin three times and assign P1=HHH, P2=HHT, P3=HTH, P4=HTT, P5=THH, P6=THT, P7=TTH, P8=TTT, but re-flip whenever the result is TTT.

Explanation: This problem involves using probability to make fair decisions among eight players, where fairness means each player has exactly 1/8 probability. To check fairness, we calculate probabilities for each option. Option A uses a fair 8-sided die with one outcome per player, giving each player exactly 1/8 probability - this is fair. Option B gives P1-P5 each 1/6 chance, P6 gets 1/6 × 1/2 = 1/12 chance, and P7-P8 each get 1/6 × 1/2 = 1/12 chance, creating numerical bias. Option C uses age, which isn't random and creates age bias. Option D's re-flipping rule when getting TTT (which would select P8) reduces P8's probability below 1/8, creating outcome bias. Remember that fairness is about equal chances for all players, not about the actual selection result.

Question 17

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 18

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 19

1. Flip a coin: if heads, Noah sits in front; if tails, flip again and let Quinn sit in front if heads and Riley sit in front if tails.
2. Spin a spinner with three equal sections labeled Noah, Quinn, and Riley. (correct answer)
3. Let the tallest person sit in front for comfort.
4. Flip a coin: heads means Noah, tails means Quinn; if Riley complains, redo the coin flip until Riley wins.

Explanation: The skill here is using probability to make fair decisions. Fairness means that each friend has an equal probability of being selected, specifically 1/3 for Noah, Quinn, and Riley. In the correct method, a spinner with three equal sections labeled for each friend is spun. This assigns equal chances because each section is the same size, so the probability of landing on any one is 1/3. One incorrect method is flipping a coin where heads picks Noah and tails leads to another flip for Quinn or Riley, introducing bias by giving Noah a probability of 1/2 and the others 1/4 each. Remember, fairness is about equal chances in each selection, not about the same friend 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 20

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.