Statistics Flashcards: Using Probability To Make Fair Decisions

What this deck covers

This deck focuses on Using Probability To Make Fair Decisions, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is $P( ext{reroll})$ in the fair $0$–$9$ method that rerolls only when $9$ occurs?

Answer: rac{1}{10}. Only digit 9 triggers reroll in the 0-9 range.

Flashcard 2: What is $P( ext{even})$ when selecting uniformly from $1$ to $10$?

Answer: rac{1}{2}. 5 even numbers (2,4,6,8,10) out of 10 total.

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

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

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

Answer: Each team has probability rac{1}{3}. Die outcomes 1,2,3 each occur with probability rac{1}{3} after rerolls.

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

Answer: Each integer has probability rac{1}{b-a+1}. Uniform means each integer gets equal probability.

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

Answer: rac{1}{12}. Equal slips ensure uniform probability for each name.

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

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

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

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

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

Answer: rac{1}{8}. Uniform selection gives each of 8 integers equal probability.

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

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

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

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

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

Answer: 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.

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

Answer: rac{3}{8}. 3 of 8 equal sectors gives probability rac{3}{8}.

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

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

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

Answer: Both are fair. Each method gives all 5 players probability rac{1}{5}.

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

Answer: P(A)=rac{2}{3} and P(B)=rac{1}{3} (not rac{1}{2} each). A gets 4 of 6 outcomes (rac{2}{3}), B gets 2 of 6 (rac{1}{3}).

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

Answer: Both are fair. Each method gives all 6 outcomes probability rac{1}{6}.

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

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

Flashcard 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?

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

Flashcard 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?

Answer: Coin flip. Coin gives rac{1}{2} each; die gives A: rac{2}{3}, B: rac{1}{3}.

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

Answer: rac{1}{5}. Rerolling 6 makes outcomes 1-5 equally likely at rac{1}{5} each.

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

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

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

Answer: rac{1}{10}. Each digit 0-9 has equal probability in uniform selection.

Flashcard 24: Which option is fair for choosing 1 of 4 people: P(A)=rac{1}{4} or P(A)=rac{1}{3}?

Answer: P(A)=rac{1}{4}. Only equal probabilities ensure fairness for all participants.

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

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

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

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

Flashcard 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?

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

Flashcard 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?

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

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

Answer: 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.

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

Answer: Each person must have probability rac{1}{n}. Equal division ensures each person has the same chance.