Welcome to the lucky booth
Story: You get some starter money and try a few luck-based games. At first they feel fair. After many rounds, a hidden pattern appears.
You start with HK$100. Four games are waiting. At first, they can all feel exciting or fair. Your job is to play, watch your notebook, and ask:
- Which booth feels nicest?
- Which booth keeps taking coins even after a lucky win?
Try the booths first. Which one feels kind? Which one quietly takes your coins?
Sample all 4 booths and play at least 8 rounds to unlock the secret math note.
One round can feel lucky. But over many rounds, each booth leaks coins on average. That average leak is called expected value. Roulette leaks about 5¢ per HK$1 play, Lucky Seven about 33¢, Scratch Card about 60¢, and Slot Machine about 49¢. The booth is not cheating. It is built so the average drifts down.
Play each booth at least once. After 8 rounds, the secret math note unlocks. Then fast-forward one booth and see what the long run does.
What does “expected value” mean?
Expected value means the average result if you played the same game again and again and again.
It does not tell you what happens on one lucky round. It tells you which way the game slowly pushes your coins over time.
Roulette (Red/Black)
American Roulette has 38 pockets: 18 red, 18 black, and 2 green (the house’s pockets).
- Win probability: 47.4%
- Payoff when you win: +HK$1
- Payoff when you lose: −HK$1
Expected value per HK$1 bet: -HK$0.05
That means every HK$1 play leaks about 5.3¢ on average.
Lucky Seven (Dice)
Roll two dice. Get a total of 7 and win HK$3 profit. Any other total loses HK$1.
- Probability of 7: 16.7% (six ways to roll a 7 out of 36 combinations)
- Payoff on 7: +HK$3
- Payoff otherwise: −HK$1
Expected value per HK$1 bet: -HK$0.33
That leaks about 33.3¢ per play.
Scratch Card
One in ten scratch cards is a winner (HK$3 prize). The rest lose HK$1.
- Win probability: 10.0%
- Payoff on win: +HK$3
- Payoff on loss: −HK$1
Expected value: -HK$0.60
That leaks about 60.0¢ per card on average.
Slot Machine
One in a hundred spins hits the jackpot (HK$50). The rest lose HK$1.
- Jackpot probability: 1.0%
- Jackpot payoff: +HK$50
- Loss payoff: −HK$1
Expected value: -HK$0.49
Even the big jackpot does not save the average. The game still leaks about 49.0¢ per spin.
Big idea: tiny edges grow over time
Every single game above has a negative expected value. That means:
- One spin might be lucky.
- Ten spins might even be profitable.
- But after lots and lots of plays, the hidden leak starts to show.
Big idea: a tiny leak can hide inside a fun game. You only notice it when the game is repeated many times.
The booth is not winning because of magic. It is winning because every game was built with a tiny average advantage.
Quiz time
In American Roulette, there are 18 red, 18 black, and 2 green pockets. What is the probability of landing on red?
There are 38 total pockets and 18 are red: 18/38 = 47.4%. Not 50% — the two green (0 and 00) pockets slightly tip the odds in the house's favour.
The Lucky Seven dice game has an expected value of -HK$0.33 per bet. What does that mean?
Expected value is an average over many plays. You might win sometimes, but over thousands of plays you'll lose about 33.3¢ per HK$1 play on average. The booth profits from this small but reliable edge.
Which of these statements best explains why casinos are profitable businesses?
When a game has a tiny edge and gets played many times, that tiny edge becomes a reliable profit.
In American Roulette, there are 38 pockets and 18 are red. If you bet HK$1 on red 38 times, how many times do you EXPECT to win? (Round to the nearest whole number.)
The expected wins = 18/38 × 38 = 18. So in 38 rounds, you'd expect to win about 18 times and lose 20 times — netting a loss of HK$2. That's where the casino's edge comes from.