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Use probability to make fair decisions, such as lotteries or random selection.

Compute participant probabilities and classify a selection method's fairness

Problem
Compute each student’s probability under the one-slip-per-student draw and compare all \(20\) probabilities to the equal-chance fairness criterion.
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Hint

Map all random outcomes—including multiple outcomes assigned to one person—before comparing probabilities.

A method is fair for people only if every participant has the intended same eventual selection probability.

Solution walkthrough

01

Map slips to students

\[20\text{ students}\longleftrightarrow20\text{ slips; one slip per student}\]

Every participant is represented by exactly one slip, so no student owns more random outcomes than another.

02

Use equal slip likelihoods

\[P(\text{any named slip})=\frac1{20}\]

The identical slips are drawn uniformly from a set of 20, making each individual slip equally likely.

03

Compute every student's chance

\[P(\text{student }i\text{ selected})=\frac1{20}\quad\text{for }i=1,\ldots,20\]

Each student's event consists of drawing that student's one slip, so all twenty participant probabilities are 1/20.

04

Apply the fairness criterion

\[\frac1{20}=\frac1{20}=\cdots=\frac1{20}\Longrightarrow\text{fair}\]

A one-winner selection is fair when all participants have equal chances. The computed probabilities satisfy that criterion and total 20(1/20)=1.

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Another way

  1. View the 20 slips as 20 equally likely elementary outcomes assigned one-to-one to the 20 students.

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Common mistake

Do not label a procedure fair merely because it is random; fairness follows only after checking that each participant owns the same total probability.

Solution walkthrough video