Using Probability to Decide Fairly

Lesson 1 of 2: Fairness and Physical Devices

In this lesson:

• Define a fair decision as equal probability
• Judge whether a procedure is fair
• Design fair decisions with a coin or die

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Define a fair decision as one giving each option equal probability
2. Judge whether a given procedure is fair or not
3. Design a fair procedure using a coin, die, or drawing lots

Five Friends, One Concert Ticket

Five friends want one concert ticket.

What counts as a fair way to decide who gets it?

Fairness Means Exactly Equal Probability

A procedure is fair when every option has the same probability of being chosen.

Clean Cases: Coin and Hat

• A fair coin decides between two people — each gets
• Identical name-slips from a well-mixed hat — each gets
• In both, every option has the same probability

Is Random the Same as Fair?

A trick coin lands heads 70% of the time. You assign Ana heads, Ben tails.

It uses chance — but is it fair? Decide before advancing.

Random Is Not the Same as Fair

The trick coin is random but not fair — Ana gets , Ben gets .

Other unfair methods: alphabetical, seniority, first to ask.

Quick Check: Fair or Not Fair?

1. Roll a fair die; even means Maya, odd means Theo
2. Pick whoever's birthday is soonest
3. Each draws an identical slip from a mixed bag

Which give equal probability?

From the Test to Building One

You can now test any procedure for fairness.

Next: construct one — split a device's equally likely outcomes into equal blocks, one per choice.

A Die Decides Fairly Among Three

Assign two faces each: 1-2, 3-4, 5-6 — every person gets .

The Same Die Handles 2 and 6

• 2 choices: 1-2-3 vs 4-6 — wait, that's uneven. Use 1-3 vs 4-6, each
• 6 choices: one face each — every choice

Unequal Grouping Is Not Fair

Faces 1-2-3 / 4-5 / 6 give — not equal, so not fair.

Your Turn: A Fair Die for Two

You have one six-sided die and two people, Lee and Sam.

Design a fair procedure and state each person's probability.

Build equal blocks before advancing.

Fair Drawing of Lots Needs Care

For lots to be fair:

• Slips must be identical in size and feel
• The container must be thoroughly mixed
• Otherwise bigger or distinct slips get grabbed more

What You Learned This Lesson

✓ Fair = equal probability — not merely "random"

✓ Build it by splitting outcomes into equal blocks

✓ A fair device with unequal blocks is still unfair

Coming Up Next: Random Generators

Computers pick fairly with random number generators — integers in a range.

Next lesson: mapping that range to choices, and the bias when the count doesn't divide evenly.