Question

a game uses a 20 - sided number solid to determine which player goes first. each face of the solid shows a unique number from 1 to 20. a group of 6 players assigns each player different numbers on the solid until all the numbers are assigned. the solid is rolled one time. the person who has the number that is rolled goes first.
the method they use is not fair for each player. which method of determining who goes first could they use instead to make it fair for each player?
assign each player different numbers on the solid until all the numbers are assigned. roll the solid 6 times. the player whose numbers occur most often goes first.
assign each player different numbers on the solid until the numbers 1 to 18 are assigned. the player who has the rolled number goes first. if 19 or 20 is rolled, start over.
assign each player a number 1 to 6. roll the solid three times and add the results. the player whose number matches the tens digit of the sum goes first. if the sum has no tens digit, start over.
assign each player a number 0 to 5. roll the solid two times and add the results. the player whose number matches the tens digit of the sum goes first. if the sum has no tens digit, the player assigned 0 goes first.

Explanation:

A fair method requires each player to have an equal probability of winning. For the second option: 18 numbers (1-18) are assigned to 6 players, so each player gets \( \frac{18}{6} = 3 \) numbers. When rolling the 20-sided solid, valid outcomes are 1-18 (probability \( \frac{18}{20} \) per roll, restarting if 19/20 are rolled). Each player’s chance is \( \frac{3}{18} = \frac{1}{6} \), which is equal for all 6 players. Other options either have unequal number assignments, depend on frequency of multiple rolls (uneven probabilities), or use sum digits with unequal likelihoods.

Answer:

Assign each player different numbers on the solid until the numbers 1 to 18 are assigned. The player who has the rolled number goes first. If 19 or 20 is rolled, start over.