Using Probabilities to Make Fair Decisions in More Complex Settings

Determine fairness of multi-stage random process flip coin then roll die; outcomes are ordered pairs.

Determine fairness of multi-stage random process spinner chooses color then die chooses number, but color sectors unequal.

Determine fairness of multi-stage random process random digit 0-9 assigned unevenly to players.

Determine fairness of multi-stage random process two fair six-sided dice are rolled; Player A wins on sum 7, Player B wins on sum 2.

Determine fairness of multi-stage random process flip a fair coin, then draw a marble from a bag with 3 red and 1 blue marble.

Determine fairness of multi-stage random process spin two fair spinners (1-4 and 1-3); Player A wins if product is even, Player B wins if product is odd.

Determine fairness of multi-stage random process draw two cards without replacement from a standard 52-card deck; Player A wins if both are hearts, Player B wins if both are clubs.

Determine fairness of multi-stage random process flip three fair coins; Player A wins with 3 heads, Player B wins with 2 heads.

Determine fairness of multi-stage random process a random integer from 0 to 99 is generated; Player A wins if 0-49, Player B wins if 50-99.

Determine fairness of multi-stage random process roll a fair six-sided die, then spin a spinner with sectors for red (1/2), blue (1/4), green (1/4).

Determine fairness of multi-stage random process draw two cards with replacement from a standard 52-card deck; Player A wins if both are aces, Player B wins if both are kings.

Determine fairness of multi-stage random process two fair six-sided dice are rolled; Player A wins if the first die is 1, Player B wins if the second die is 6.

Determine fairness of multi-stage random process two numbers are chosen without replacement from {1, 2, 3, 4}; Player A wins if sum is 3, Player B wins if sum is 5.

Determine fairness of multi-stage random process flip a fair coin, then roll a loaded die where 6 is twice as likely as any other number.

Design fair selection with unequal groups choose one of 3 clubs with sizes 10,20,30 fairly by person.

Design fair selection with unequal groups use random digits for 6 people.

Design fair selection with unequal groups pick winner among 12 people using two dice sums.

Design fair selection with unequal groups groups have unequal sizes but each group should have equal chance.

Design fair selection with unequal groups choose one student from a school with 4 grades of sizes 100, 120, 110, 90 fairly by student.

Design fair selection with unequal groups make a fair 50/50 choice using a coin that lands heads 60% of the time.

Design fair selection with unequal groups select one item from 3 categories with 5, 8, and 10 items respectively, ensuring each category has an equal chance of being selected.

Design fair selection with unequal groups select one of 4 options fairly using a standard six-sided die.

Design fair selection with unequal groups choose one student from Class A (25 students) and Class B (30 students) fairly by student.

Design fair selection with unequal groups select one of 7 participants fairly using a 10-sided die.

Design fair selection with unequal groups choose one project from Department X (5 projects), Department Y (7 projects), and Department Z (4 projects), ensuring each department has an equal chance of having its project chosen.

Evaluate fairness with conditional probabilities in two cards are drawn without replacement and different players win on different ordered pairs.

Evaluate fairness with conditional probabilities in a branch game gives Player A one branch with probability 1/2 and Player B two branches with total probability 1/2.

Evaluate fairness with conditional probabilities in a coin decides which spinner is used, and the spinners have different color distributions.

Evaluate fairness with conditional probabilities in drawing two cards from a standard deck without replacement; Player A wins if both are aces, Player B wins if the first is a king and the second is a queen.

Evaluate fairness with conditional probabilities in an urn contains 5 red and 5 blue marbles; two are drawn without replacement; Player A wins if the first is red, Player B wins if the second is red.

Evaluate fairness with conditional probabilities in a game show where a contestant picks one of three doors, then a non-chosen goat door is opened, and the contestant can switch or stick.

Evaluate fairness with conditional probabilities in a batch of 10 items has 2 defects; Inspector A checks the first item, Inspector B checks the second item if the first was good.

Evaluate fairness with conditional probabilities in drawing two names from a hat containing 3 boys and 2 girls; Player A wins if the first name is a boy, Player B wins if the second name is a girl.

Evaluate fairness with conditional probabilities in a die roll determines if you draw from Deck A (even) or Deck B (odd); Deck A has 3 red, 2 blue; Deck B has 1 red, 4 blue; Player A wins on red, Player B wins on blue.

Evaluate fairness with conditional probabilities in a committee of 2 is chosen from 4 men and 3 women; Player A wins if both are men, Player B wins if the first is a woman and the second is a man.

Evaluate fairness with conditional probabilities in a coin flip determines which bag to draw from; Bag 1 has 4 green, 1 yellow; Bag 2 has 2 green, 3 yellow; Player A wins on green, Player B wins on yellow.

Evaluate fairness with combinatorial sample spaces for choosing a 2-person committee from 5 students where one student is favored if included.

Evaluate fairness with combinatorial sample spaces for drawing 3 names from a class where each team has different size.

Evaluate fairness with combinatorial sample spaces for assigning prizes by unordered card pairs.

Evaluate fairness with combinatorial sample spaces for forming teams from unequal groups.

Evaluate fairness with combinatorial sample spaces for choosing a president and a vice-president from 6 candidates.

Evaluate fairness with combinatorial sample spaces for a lottery where 3 winners are drawn from 50 tickets, and one person bought 5 tickets while another bought 10.

Evaluate fairness with combinatorial sample spaces for drawing 2 marbles from a bag containing 3 red and 4 blue marbles, to see if drawing two of the same color is fair.

Evaluate fairness with combinatorial sample spaces for assigning 3 specific people to 3 seats out of 5 available seats in a row.

Evaluate fairness with combinatorial sample spaces for drawing 5 cards from a standard deck to determine if getting a flush is a fair outcome compared to other hands.

Evaluate fairness with combinatorial sample spaces for selecting 2 representatives from a group of 10 employees, 2 of whom are senior staff and 8 are junior staff.

Evaluate fairness with combinatorial sample spaces for forming a 3-person committee from 4 doctors and 3 nurses, where at least one doctor must be included.

Use simulation results to assess fairness of a board game spinner and die determine winners; simulation gives A: 4980, B: 5020 in 10000 trials.

Use simulation results to assess fairness of a drawing process gives A: 6200, B: 3800 in 10000 trials.

Use simulation results to assess fairness of three strategies simulated 5000 times produce win rates 0.34, 0.33, 0.33.

Use simulation results to assess fairness of a complex game has small-sample simulation counts 18 and 12 in 30 trials.

Use simulation results to assess fairness of a coin flip simulation yields 49995 heads and 50005 tails in 100000 trials.

Use simulation results to assess fairness of a card drawing process is simulated 20000 times, resulting in 7000 draws for event A and 13000 for event B.

Use simulation results to assess fairness of a dice roll simulation shows 1050 rolls of '6' and 8950 rolls of other numbers in 10000 trials.

Use simulation results to assess fairness of a game with two players, A and B, is simulated 50 times, with A winning 30 times and B winning 20 times.

Use simulation results to assess fairness of a lottery simulation with three outcomes (Win, Lose, Draw) in 100000 trials shows 33000 Wins, 33500 Loses, and 33500 Draws.

Use simulation results to assess fairness of a random number generator produces 1200 even numbers and 800 odd numbers in 2000 trials.

Use simulation results to assess fairness of a voting system simulation with candidates X, Y, Z in 15000 trials yields X: 4800, Y: 5100, Z: 5100 votes.

Use simulation results to assess fairness of a two-sided coin is flipped 200 times, resulting in 120 heads and 80 tails.

Use simulation results to assess fairness of a complex algorithm's output is simulated 500000 times, with success rate 0.4999 and failure rate 0.5001.

Use simulation results to assess fairness of a prize drawing simulation in 100 trials gives 60 wins and 40 losses.

Modify biased process random digits 0-9 assign 0-3 to A and 4-9 to B to make it fair.

Modify biased process two dice sums 2-12 are assigned directly to players to make it fair.

Modify biased process spinner sectors are 20%,30%,50% for three players to make it fair.

Modify biased process a bag has unequal color counts assigned to different players to make it fair.

Modify biased process a weighted coin lands on heads 60% of the time and tails 40%, with heads assigned to Player A and tails to Player B to make it fair.

Modify biased process a random integer from 1 to 100 is generated, with 1-40 assigned to A, 41-70 to B, and 71-100 to C to make it fair.

Modify biased process a committee of 7 men and 3 women randomly selects one member to be chairperson, with men assigned to Player A and women to Player B to make it fair.

Modify biased process a six-sided die is rolled; Player A wins on 1 or 2, Player B wins on 3, 4, or 5, and Player C wins on 6 to make it fair.

Modify biased process a person is chosen from a group where 60% are right-handed and 40% are left-handed; Player A wins if a right-handed person is chosen, Player B wins if a left-handed person is chosen to make it fair.

Modify biased process a lottery has 100 tickets; Player A buys 50, Player B buys 30, and Player C buys 20; one ticket is drawn randomly to make it fair.

Modify biased process a bag contains 5 red balls, 3 blue balls, and 2 green balls; Player A wins if red, Player B if blue, Player C if green to make it fair.

Compare fairness and practicality of decision methods Method A uses rejection sampling with many rerolls; Method B uses a uniform random number list.

Compare fairness and practicality of decision methods drawing names from identical slips versus letting volunteers enter multiple times.

Compare fairness and practicality of decision methods using a phone randomizer versus two dice sums for 11 people.

Compare fairness and practicality of decision methods drawing a name from a hat with 10 slips versus being the first to guess a number between 1 and 100.

Compare fairness and practicality of decision methods assigning a task based on who finished their last task fastest versus using a spinner with 5 equal sections.

Compare fairness and practicality of decision methods picking the first 20 students who arrive at school versus using a random number generator to select 20 student IDs.

Compare fairness and practicality of decision methods flipping a coin versus playing Rock-Paper-Scissors to decide who goes first in a game.

Compare fairness and practicality of decision methods allocating limited resources on a first-come, first-served basis versus a lottery system where each applicant gets one entry.

Compare fairness and practicality of decision methods asking for volunteers versus drawing names from cards to choose a representative from a group.

Compare fairness and practicality of decision methods alternating assignment (A, B, A, B) versus each student rolling a die (even to A, odd to B) for group assignment.

Compare fairness and practicality of decision methods picking a day of the week based on personal preference versus using a spinner with 7 equal sections.

Identify hidden assumptions in fairness model a dice game assumes each sum from 2 to 12 is equally likely.

Identify hidden assumptions in fairness model a spinner is treated as fair because it has three colors.

Identify hidden assumptions in fairness model a raffle assumes each person has one ticket.

Identify hidden assumptions in fairness model two draws are multiplied as independent after the first item is not replaced.

Identify hidden assumptions in fairness model a game uses a standard six-sided die and assumes each number from 1 to 6 has an equal chance of appearing.

Identify hidden assumptions in fairness model a random selection process from two groups is assumed fair because one person is chosen from each group.

Identify hidden assumptions in fairness model a teacher picks a student's name from a hat, assuming each student has an equal chance of being chosen.

Identify hidden assumptions in fairness model the probability of selecting a red card, then another red card, from a standard deck is calculated by P(Red) * P(Red).

Identify hidden assumptions in fairness model the probability of rain on both Saturday and Sunday is found by multiplying the probability of rain on Saturday by the probability of rain on Sunday.

Identify hidden assumptions in fairness model the overall reliability of a system with two components is found by multiplying the reliability of each component, assuming they operate independently.

Identify hidden assumptions in fairness model a political poll assumes that every voter in the population has an equal chance of being contacted and responding.

Explain fairness using probability evidence for Player A wins on 4 of 12 equally likely outcomes and Player B wins on 8 of 12.

Explain fairness using probability evidence for two players have simulated win rates 0.51 and 0.49 over 20000 trials.

Explain fairness using probability evidence for groups have equal chance but individuals in smaller groups have larger chance.

Explain fairness using probability evidence for a spinner has sections labeled Red, Blue, Green, where Red is 90 degrees, Blue is 180 degrees, and Green is 90 degrees, and players choose colors.

Explain fairness using probability evidence for a raffle where John has 5 tickets and Mary has 10 tickets out of a total of 20 tickets.

Explain fairness using probability evidence for a game where Player A rolls a six-sided die and wins if they roll an even number, and Player B rolls a four-sided die and wins if they roll an even number.

Explain fairness using probability evidence for a game where two teams played 1000 matches, and Team A won 498 times and Team B won 502 times.

Explain fairness using probability evidence for a new card game was simulated 10000 times, showing Player 1 winning 65% of the time and Player 2 winning 35% of the time.

Explain fairness using probability evidence for a drawing where three departments each submit 10 entries for a prize, but one department has 5 employees and another has 10 employees.

Explain fairness using probability evidence for a coin is flipped 100 times and lands on heads 70 times and tails 30 times.

Explain fairness using probability evidence for a game where Player A wins if they draw a red card from a standard deck, and Player B wins if they roll an even number on a six-sided die.

Explain fairness using probability evidence for a lottery where choosing numbers 1-5 has a 1/100 chance of winning, but choosing numbers 6-10 has a 1/50 chance of winning.

Explain fairness using probability evidence for a bag contains 3 red and 3 blue balls; a ball is drawn, its color noted, and then replaced, and then another ball is drawn. Player R wins if both are red, Player B wins if both are blue.

Explain fairness using probability evidence for rolling two standard six-sided dice, where Player A wins if the sum is 7, and Player B wins if the sum is 2 or 12.

Resolve fairness dispute one student wants to use dice sums for 11 people; another wants random digits with rejection.

Resolve fairness dispute a game assigns more winning cards to one player because that player goes second.

Resolve fairness dispute a classroom contest can choose uniformly by team or uniformly by student.

Resolve fairness dispute two methods are both fair but one requires many rejected outcomes.

Resolve fairness dispute a lottery drawing uses names on slips of paper, but some slips are larger than others; an alternative is a random number generator.

Resolve fairness dispute a card game uses a shuffled deck, but one player suspects the shuffling method is not random, suggesting a computer shuffle instead.

Resolve fairness dispute a tournament director wants to seed players; one method uses past performance rankings, another uses a completely random draw.

Resolve fairness dispute a raffle has participants with varying numbers of tickets; one participant argues for an equal chance per person, another for an equal chance per ticket.

Resolve fairness dispute a teacher needs to form project groups; one method is random assignment, another is allowing students to self-select.

Resolve fairness dispute two people need to decide who goes first in a game; one suggests using a coin known to be biased, the other suggests rolling a standard die and assigning outcomes.

Resolve fairness dispute a team needs to assign an undesirable task; one member proposes drawing names from a hat, another proposes assigning it to the newest team member.

Identify when fairness conflicts with other goals in equal chance by school gives students in small schools higher individual probability.

Identify when fairness conflicts with other goals in strictly fair lottery may take many rerolls.

Identify when fairness conflicts with other goals in weighted lottery gives needier applicants more entries.

Identify when fairness conflicts with other goals in random assignment balances probability but may not guarantee every subgroup is represented.

Identify when fairness conflicts with other goals in using a truly random number generator for a small decision takes more time than a quick, less rigorous method.

Identify when fairness conflicts with other goals in randomly selecting students for a school assembly panel may not ensure a balanced representation of grade levels.

Identify when fairness conflicts with other goals in a lottery for limited medical resources gives equal probability to all patients regardless of severity of illness.

Identify when fairness conflicts with other goals in a completely random assignment of tasks might lead to some tasks being assigned to unqualified personnel, requiring more time to fix.

Identify when fairness conflicts with other goals in randomly selecting members for a cultural event planning committee might not include members from all relevant cultural groups.

Identify when fairness conflicts with other goals in a lottery for scholarship funds gives every applicant an equal chance, even if some have significantly greater financial need.

Identify when fairness conflicts with other goals in randomly selecting participants for a scientific study might result in an unrepresentative sample for specific demographic analysis.

Correct complex fairness-analysis error The game is fair because each player has the same number of dice sums.

Correct complex fairness-analysis error The process is fair because the tree has the same number of branches for each player.

Correct complex fairness-analysis error Simulation showed 56 wins out of 100, so the process is definitely unfair.

Correct complex fairness-analysis error Without replacement can be treated as independent because the first draw was random.

Correct complex fairness-analysis error When rolling two standard dice, there are 11 possible sums (2 through 12), so the probability of rolling a 7 is 1/11.

Correct complex fairness-analysis error Since the last three coin flips were heads, the next flip is more likely to be tails to balance it out.

Correct complex fairness-analysis error I simulated a 1/4 chance by rolling a die and counting 1 or 2 as a win, and 3-6 as a loss.

Correct complex fairness-analysis error A game where you pay $1 to play and win $2 if a coin lands heads is fair because you can win money.

Correct complex fairness-analysis error When drawing 3 cards from a deck, the number of possible hands is 52 * 51 * 50 because the order matters.

Correct complex fairness-analysis error If a medical test is 99% accurate (P(positive|disease) = 0.99), and you test positive, you almost certainly have the disease.

Correct complex fairness-analysis error When flipping two coins, the possible outcomes are HH, HT, TT, so the probability of getting one head and one tail is 1/3.

Correct complex fairness-analysis error After 10 coin flips, I got 7 heads and 3 tails, so the coin is biased towards heads.

Correct complex fairness-analysis error The probability of getting at least one head in two coin flips is 1/2 because one of the two flips could be heads.

Correct complex fairness-analysis error I feel lucky today, so the game is fair for me.