Time to switch sides
Story: In the last lesson you played the booths and watched your coins leak away. Today the booth is YOURS. You set the prize. Your job: make a game people want to play — that still earns you coins by the end of the day.
Your booth is called Roll-a-Six. The rules are dead simple:
- A customer pays you HK$1 and rolls one die.
- If they roll a 6 (a 16.7% chance), you pay them a prize.
- Any other number — you keep their HK$1.
The only decision you make is the prize. Set it too small and nobody feels like playing. Set it too big and you become the one leaking coins.
🎪 Your booth: Roll-a-Six
Each customer pays you HK$1 and rolls one die. If they roll a 6, you pay them the prize. You choose the prize:
Look at the two averages. With only a few customers they can be far apart — that's luck being loud. The more customers you serve, the closer your actual average creeps to the math's prediction. A booth owner doesn't need luck; they just need a prize slightly below the fair one, and lots of customers.
Try this experiment: serve 1 customer at a time and watch the two averages. Then fast-forward 500 customers. Do the same at a HK$5 prize, and at HK$6. When does your actual average hug the prediction?
The fair prize — where nobody has an edge
Out of every 6 rolls, on average 1 is a six and 5 are not. So for the game to be perfectly fair, one win should cancel out 5 losses:
- You collect HK$1 × 5 = HK$5 from the non-sixes.
- You pay out the prize once.
If the prize is exactly HK$5, everything cancels. The player’s expected value is +HK$0.00 per play — a truly fair game. Nobody drifts up, nobody drifts down; only luck moves the coins.
The house edge — shave the prize
Now do what every real booth does: pay a little less than the fair prize.
At a HK$4 prize:
- Player’s expected value: -HK$0.17 per play
- Your average earnings: +HK$0.17 per customer
About 17 cents per customer. That sounds like nothing — no customer would even notice it. One lucky player can still walk away a winner. But you aren’t playing once. You’re playing every round, against everyone.
Big idea: the house doesn’t beat the players. The house beats luck itself — by playing so many rounds that the tiny average edge has nowhere to hide.
Watch what volume does to a 17-cent edge:
- A quiet Market Day, 300 customers: expect about HK$50 profit.
- A busy weekend, 3000 customers: expect about HK$500.
Same tiny edge, more repetitions, bigger and more reliable total. That is the entire business model of every casino on Earth.
And it works in reverse: at a HK$6 prize your “average earnings” become -HK$0.17 per customer — now the players own the edge, and a busy day just makes you broke faster.
Back to the booths from Lesson 6
Remember Roulette? Its expected value was -HK$0.05 per HK$1 bet — a leak so small it feels invisible. But after 1000 spins, the casino expects about HK$53 of your money. Not by cheating. Not by luck. By the exact same trick you just used at your own booth: a prize slightly below fair, repeated many times.
Quiz time
Your booth charges HK$1 per roll and pays a prize on a six. What prize makes the game perfectly fair?
Each six comes with about 5 non-sixes. Those 5 losses hand you HK$5 — so the prize must be HK$5 to cancel them exactly. That's why the expected value at HK$5 is +HK$0.00.
At a HK$4 prize your edge is only about 17 cents per customer. Why is that enough to run a business on?
One customer might win big — that's fine. Over 300 customers the tiny edge adds up to about HK$50, and over 3000 customers about HK$500. Volume turns a whisper of an edge into a steady income.
A friend says: 'I won HK$50 at the casino, so the house edge must be fake.' What's the best reply?
Expected value says nothing about one visit. It says what the average drifts toward over thousands of plays — and the casino is on the other side of ALL of those plays at once.
You set the prize to HK$4 and serve 300 customers on Market Day. About how many HK$ profit should you EXPECT? (Round to the nearest whole number.)
Your edge is +HK$0.17 per customer, so 300 customers × that edge ≈ HK$50. Any single day can land above or below — but this is the number the days will average around.
For your Casino Night speech
You have to explain why the house wins over time in three sentences. Here’s the frame — fill it with your own words:
- Every game pays winners a bit less than the fair prize (name your booth’s numbers!).
- That makes the average result a small leak from player to house — too small to feel in one game.
- The house plays thousands of rounds, so the average takes over and the leak becomes steady profit.