Fair random choice guides — page 2 of 4

A dice pool counts individual dice that meet a stated threshold. If 6 fair d10 dice succeed on 8, 9, or 10, one die has probability 3/10 and exactly 2 successes have probability C(6,2)(0.3)^2(0.7)^4 = 32.41%.

A d100 roll and percentile dice can both represent 100 equally likely values, but the notation must be fixed. In the common percentile convention, a tens die of 40 and a ones die of 7 means 47, while 00 and 0 means 100.

Drawing with replacement returns the selected item before the next draw, so probabilities reset and repeats are possible. Drawing without replacement removes the item, so a five-name pool has 4 names after the first draw and cannot select the same winner twice.

To pick multiple winners without repeats, freeze one eligible list, make one uniform shuffle, and take the first k distinct entries. From 12 eligible names, choosing 3 winners without replacement gives each named person a 3/12 = 25% chance of appearing somewhere in the winner list.

A fair shuffle gives every complete ordering of n distinct list entries the same probability. For 4 names there are 4! = 24 possible orders, so one named order has probability 1/24; a correct Fisher–Yates shuffle selects one of those orders without keeping an early-entry advantage.

Choose a shuffle when every item needs one position in a complete no-repeat order; choose a one-item picker when only one result is needed and later picks may be independent. A shuffled list of 8 names has 8 positions, while one pick returns one name with probability 1/8.

Sorting a list with a comparator that returns random positive or negative values is not a reliable shuffle because sorting algorithms expect a consistent comparator. Use Fisher–Yates instead: for index i, choose one uniformly random index from 0 through i and swap once.

To create random pairs from one list, shuffle the eligible people once and group adjacent positions. Eight people create 4 pairs; if 7 people are eligible, the result is 3 pairs and 1 announced leftover, so the leftover rule must be set before the shuffle.

To create random pairs from two lists, verify the roles and counts, shuffle one list, and match it position by position to the other. With 5 mentors and 5 learners, each named mentor has probability 1/5 of being matched to any particular learner if the learner list is shuffled uniformly.

To randomly assign people to equal-sized groups, shuffle the confirmed roster once and split the order into equal consecutive blocks. With 24 people and groups of 6, the first 6 shuffled names form group 1 through the final 6 names in group 4; each group has exactly 6 members.

Random teams give every eligible person an equal chance of a group position; balanced teams deliberately use stated ability data to make group totals closer. With 12 players split into 3 teams of 4, random assignment guarantees four players per team, while balance requires a declared measure such as ratings or experience.

A fair random classroom seating plan starts with the usable seats and the eligible students, records accommodations and fixed restrictions before the draw, then assigns the remaining students to the remaining seats once. In a room with 24 usable seats and 24 eligible students, each unrestricted student has a 1/24 chance of a specified unrestricted seat before any constraints are applied.

Choose a verifiable random speaking order by freezing the roster, producing one complete shuffle, numbering every position, and keeping the first output. For 8 listed speakers, a named person has probability 1/8 of position 1 and exactly one position from 1 through 8; a sequence of independent name picks does not provide that no-repeat guarantee.

A chore wheel is fair only after the household defines the period, eligible people, eligible chores, exclusions, and what happens after a completed or declined task. If 4 housemates rotate through 4 equally sized weekly chores, a one-to-one shuffle gives each person one task; equal probability does not make unequal effort equal.

Run Secret Santa by creating one complete giver-to-recipient assignment, rejecting self-matches and every stated exclusion before anyone receives a name, then privately delivering the final pairs. With 6 participants, a valid draw has six givers and six different recipients; it is not six independent name picks.

A random name picker selects one entry from a list; a Secret Santa draw creates a complete one-to-one assignment in which every participant gives once and receives once. With 5 people, one picker output has 5 possible winners, while a self-free Secret Santa assignment has !5 = 44 valid complete maps before additional exclusions.

Seed a tournament bracket fairly by deciding whether the event rewards prior results, uses a random draw, or combines fixed seed bands with a random draw inside each band before the bracket is published. In an 8-player single-elimination bracket, 7 matches determine one winner; the placement rule changes who can meet before the final.

Choose round robin when each entrant should play every other entrant and time permits; choose single elimination when a quick winner matters more than guaranteed matches. With 8 entrants, a single-elimination bracket needs 7 matches, while one round robin needs 8 × 7 ÷ 2 = 28 matches.

A round-robin schedule gives every entrant one match against every other entrant. For n entrants, the number of matches is n(n−1)/2: 6 teams need 6 × 5 ÷ 2 = 15 matches. With an even number of teams, each round contains n/2 matches and there are n−1 rounds.

Make a weighted random choice fair by naming every option, publishing each positive weight and the total, mapping weight to probability before the draw, and retaining the first result. With weights 6, 3, and 1, the total is 10, so the three options have probabilities 60%, 30%, and 10% respectively; random does not mean equal.