Back to Exercise: Use probabilities to make fair decisions

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

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Warm-Up: What Makes a Decision Fair

These problems review the meaning of a fair decision.

A decision procedure between several people is called fair. Which statement is the correct definition of a fair procedure?

Every person has exactly the same probability of being chosen.

The procedure uses chance in some way, no matter the probabilities.

The person who asks first is chosen.

The procedure always produces a different winner each time.

A teacher must choose one of two students for a prize. Which method gives each student probability exactly $\frac{1}{2}$ ?

Flip a fair coin: heads for one student, tails for the other.

Choose the student whose name comes first alphabetically.

Choose the student who has been in the class longer.

Choose whichever student raises a hand first.

A fair six-sided die is used to choose fairly among 3 people. Two faces are assigned to each person: person X gets $\{1, 2\}$ , person Y gets $\{3, 4\}$ , person Z gets $\{5, 6\}$ . What is each person's probability of being chosen?

Each person has probability $\frac{2}{6} = \frac{1}{3}$ .

Each person has probability $\frac{1}{6}$ , one per face.

The procedure is unfair because the faces have different numbers.

Person Z has the highest probability because $\{5, 6\}$ are the largest faces.

Fluency Practice

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

A coin is biased: it lands heads $70\%$ of the time and tails $30\%$ of the time. It is used to decide between two people (heads for one, tails for the other). Is this procedure fair?

No — it uses chance but the two outcomes are not equally likely, so it is not fair.

Yes — any coin flip is fair because it is random.

Yes — heads and tails are still two different outcomes.

It cannot be decided without flipping the coin many times.

You want to choose fairly among 6 people using one fair six-sided die, assigning one face to each person. What is each person's probability of being chosen? Enter your answer as a fraction.

A random number generator produces integers from 1 to 100, each equally likely. To choose between two people, the blocks are: person P gets $1$ – $50$ , person Q gets $51$ – $100$ . Is this procedure fair?

Yes — each block has 50 of the 100 equally likely integers, so each probability is $\frac{50}{100} = \frac{1}{2}$ .

No — a computer generator can never be trusted to be fair.

No — the numbers $51$ – $100$ are larger, so person Q is favored.

It is fair only if you also flip a coin afterward.

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.

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