Exercises: Use Probabilities to Make Fair Decisions

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.

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.

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.

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.

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.

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.