Back to Use probabilities to make fair decisions — Problem 4 · Task Set 33

Exercises: Use Probabilities to Make Fair Decisions

Work through each section in order. A decision procedure is FAIR when every
option has exactly the same probability of being chosen. For design problems,
describe the mapping and state each option's probability. When a device's
outcomes do not divide evenly among the choices, use a divisible range or
rejection sampling (reject the leftover outcomes and redraw), and verify each
choice has probability $\frac{1}{k}$ .

Grade 11·19 problems·~35 min·Common Core Math - HS Statistics and Probability·group·hss-md-b-6

Work through problems with immediate feedback

Fluency Practice

Judge whether each procedure is fair and design simple fair mappings.

A fair six-sided die is used to choose among 3 people, but the faces are assigned unequally: person X gets $\{1, 2, 3\}$ , person Y gets $\{4, 5\}$ , person Z gets $\{6\}$ . What is wrong, and how can it be fixed?

The blocks are unequal ( $\frac{3}{6}, \frac{2}{6}, \frac{1}{6}$ ), so it is unfair. Fix it by giving each person two faces: $\{1,2\}, \{3,4\}, \{5,6\}$ , each $\frac{1}{3}$ .

Nothing is wrong; every face is assigned, so it is fair.

Person Z is favored because $\{6\}$ is a single, special face.

It is unfair, and the only fix is to buy a three-sided die.