Using Probability to Make Fair Decisions - Statistics and Probability

1. What is $P(\text{reroll})$ in the fair $0$–$9$ method that rerolls only when $9$ occurs?

   $\frac{1}{10}$. Only digit 9 triggers reroll in the 0-9 range.

2. What is $P(\text{even})$ when selecting uniformly from $1$ to $10$?

   $\frac{1}{2}$. 5 even numbers (2,4,6,8,10) out of 10 total.

3. What is the definition of a fair decision procedure in terms of probabilities?

   Each outcome has the intended equal probability (often $\frac{1}{n}$). Fair means all participants have the same chance of being selected.

4. Identify the probability each team gets if 3 teams use reroll-on-$4,5,6$ with a fair die.

   Each team has probability $\frac{1}{3}$. Die outcomes 1,2,3 each occur with probability $\frac{1}{3}$ after rerolls.

5. What is the definition of a uniform random number generator on integers $a$ through $b$?

   Each integer has probability $\frac{1}{b-a+1}$. Uniform means each integer gets equal probability.

6. What is the probability of drawing a specific name when 12 equal slips are mixed and one is drawn?

   $\frac{1}{12}$. Equal slips ensure uniform probability for each name.

7. Which random-number set is best to pick 1 of 5 people fairly: $1,2,3,4,5$ or $1,2,3,4$?

   Use $1,2,3,4,5$ (one number per person). Need exactly 5 numbers to assign one per person.

8. Find a fair rule for 3 people using a fair die by allowing rerolls on some outcomes.

   Use $1$-$3$ for A,B,C and reroll on $4$-$6$. Rerolling unwanted outcomes maintains equal probabilities.

9. What is $P(1)$ when selecting uniformly from integers $1$ through $8$?

   $\frac{1}{8}$. Uniform selection gives each of 8 integers equal probability.

10. What condition must a mapping from equally likely outcomes to people satisfy to be fair?

    Each person must be assigned the same number of equally likely outcomes. Equal outcome counts ensure equal probabilities.

11. Identify the fair way to use a fair coin to choose between 3 options using two flips.

    Use $HH,HT,TH$ for options and reroll on $TT$. Three outcomes ($HH$, $HT$, $TH$) map to three options.

12. Identify the fair method to pick 1 of 3 options using a random digit $0$–$9$ without bias.

    Use $0$–$8$; map $0$–$2$, $3$–$5$, $6$–$8$; reroll on $9$. Assigns 3 digits to each option, rerolls the 10th digit for fairness.

13. What is $P(A)$ if a spinner has 8 equal sectors and 3 are labeled $A$?

    $\frac{3}{8}$. 3 of 8 equal sectors gives probability $\frac{3}{8}$.

14. Which option is fair for 8 people: use RNG $1$-$8$ once, or use RNG $1$-$10$ once without rerolls?

    Use RNG $1$-$8$ once (or reroll $9$-$10$). RNG $1$-$8$ gives each person exactly one number.

15. Which option is fair for 5 players: numbers $1$–$5$ on equal slips, or a fair die with reroll on $6$?

    Both are fair. Each method gives all 5 players probability $\frac{1}{5}$.

16. Find the probability difference from fairness: die rule $1$–$4$ for A, $5$–$6$ for B (fairness target $\frac{1}{2}$).

    $P(A)=\frac{2}{3}$ and $P(B)=\frac{1}{3}$ (not $\frac{1}{2}$ each). A gets 4 of 6 outcomes ($\frac{2}{3}$), B gets 2 of 6 ($\frac{1}{3}$).

17. Which option is fair for 6 outcomes: one roll of a fair die, or drawing from 6 equal slips labeled $1$–$6$?

    Both are fair. Each method gives all 6 outcomes probability $\frac{1}{6}$.

18. Which method is fair for 3 people: roll a fair die once, or flip a fair coin once?

    Roll a fair die once (coin has only $2$ outcomes). A die has 6 outcomes, allowing fair assignment to 3 people.

19. What is the fairness error if 2 people flip a coin but one wins on heads or tails and the other only on heads?

    Not fair; probabilities are $\frac{1}{2}$ vs $1$. One person wins always while the other wins only half the time.

20. Which option is fair for choosing between 2 people: coin flip, or rolling a die where $1$–$4$ is A and $5$–$6$ is B?

    Coin flip. Coin gives $\frac{1}{2}$ each; die gives A: $\frac{2}{3}$, B: $\frac{1}{3}$.

21. What is $P(\text{player 3})$ if a fair die assigns players $1$–$5$ and rerolls on $6$?

    $\frac{1}{5}$. Rerolling 6 makes outcomes 1-5 equally likely at $\frac{1}{5}$ each.

22. Identify the fair spinner: (1) $6$ equal sectors, (2) $4$ equal and $2$ double-width sectors.

    Spinner (1) with $6$ equal sectors. Equal sectors give each outcome the same probability.

23. What is $P(7)$ when a random number generator picks uniformly from $0$ to $9$?

    $\frac{1}{10}$. Each digit 0-9 has equal probability in uniform selection.

24. Which option is fair for choosing 1 of 4 people: $P(A)=\frac{1}{4}$ or $P(A)=\frac{1}{3}$?

    $P(A)=\frac{1}{4}$. Only equal probabilities ensure fairness for all participants.

25. Which option makes the decision fair: choose A with $0.5$ or choose A with $0.6$ (two outcomes)?

    Choose A with $0.5$ (and B with $0.5$). Equal probabilities ($0.5$ each) ensure fairness.

26. Identify the problem: RNG gives $1$-$10$ but you assign A:$1$-$3$, B:$4$-$6$, C:$7$-$10$.

    Not fair; A and B get $3$ numbers, C gets $4$ numbers. C has higher probability ($\frac{4}{10}$) than A or B ($\frac{3}{10}$).

27. Which option is fair for 4 players: random integer $1$–$20$ with 5 numbers each, or random integer $1$–$18$ with 4 or 5 numbers each?

    Random integer $1$–$20$ with 5 numbers each. Equal assignment (5 each) ensures $\frac{1}{4}$ probability per player.

28. Which option is fair for 3 teams: roll a fair die and reroll on $4,5,6$, or spin a 4-equal-section spinner?

    Roll a fair die and reroll on $4,5,6$. Rerolling 4,5,6 gives outcomes 1,2,3 equal $\frac{1}{3}$ probability each.

29. What is the fair mapping if a random number generator gives integers $1$ to $20$ and you must choose 4 people?

    Assign 5 numbers to each person (e.g., $1$-$5$, $6$-$10$, $11$-$15$, $16$-$20$). Each person gets $\frac{20}{4} = 5$ numbers for equal probability.

30. What probability must each of $n$ people have to win in a perfectly fair random selection?

    Each person must have probability $\frac{1}{n}$. Equal division ensures each person has the same chance.

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