By the end of this lesson, learners should be able to:
This lesson uses fictional money for probability and programming education—not real-money betting.
A game of chance is mathematically fair if neither player has an expected long-run advantage.
If a player repeatedly makes the same bet, their average net gain should approach zero over a very large number of plays:
\[ \text{Expected value for player} = 0 \]
For a game with possible outcomes \(i\):
\[ EV = \sum_i P(i) \times \text{net payoff}(i) \]
A fair game has:
\[ EV = 0 \]
This does not mean that every player breaks even in a single session. A player may win ₹500 in one round and lose ₹500 in the next. Fairness concerns the average outcome over many repetitions.
A casino may describe a game as “fair” when it is honestly run: the rules are clear, outcomes are random, and payouts match the published schedule. But the game is usually not mathematically fair for the player.
| Type of fairness | Meaning | Expected player result |
|---|---|---|
| Mathematical fairness | Payout exactly matches the probability of winning | ₹0 in the long run |
| Casino or operational fairness | Random, transparent, and follows advertised rules | Usually a long-run loss |
| House-edge game | Payout is slightly below fair value | Negative expected value |
The casino’s advantage is called the house edge:
\[ \text{House edge} = -\frac{\text{Player expected value}}{\text{Stake}} \]
If the house edge is 5%, then on average the player loses ₹5 per ₹100 wagered over a very large number of plays.
The related measure is Return to Player (RTP):
\[ \text{RTP} = 1 - \text{House edge} \]
So, a 5% house edge means:
\[ \text{RTP} = 95\% \]
Suppose:
Then:
\[ G = \frac{b(1-h)}{p} \]
For a mathematically fair game, set \(h=0\):
\[ G = \frac{b}{p} \]
For a casino version, choose a positive value of \(h\):
\[ G = \frac{b(1-h)}{p} \]
The player’s net profit after a winning round is:
\[ \text{Net win} = G-b \]
A player stakes ₹100 and chooses one number from 1 to 6. One die is rolled.
The probability of winning is:
\[ p = \frac{1}{6} \]
\[ G = \frac{100}{1/6} = \text{₹600} \]
The ₹600 is the gross payout, including the original stake. Therefore:
\[ \text{Net win} = \text{₹600} - \text{₹100} = \text{₹500} \]
| Result | Probability | Player net result |
|---|---|---|
| Selected face appears | \(1/6\) | +₹500 |
| Any other face appears | \(5/6\) | -₹100 |
\[ EV = \left(\frac{1}{6}\times500\right) + \left(\frac{5}{6}\times-100\right) = \text{₹0} \]
So this is mathematically fair.
Set \(h=0.05\):
\[ G = \frac{100(1-0.05)}{1/6} = \text{₹570} \]
The player earns a net ₹470 after a win, rather than ₹500.
| Result | Probability | Player net result |
|---|---|---|
| Selected face appears | \(1/6\) | +₹470 |
| Any other face appears | \(5/6\) | -₹100 |
\[ EV = \left(\frac{1}{6}\times470\right) + \left(\frac{5}{6}\times-100\right) = -\text{₹5} \]
The expected player loss is ₹5 per ₹100 bet, which is a 5% house edge.
When two dice are rolled, their sums do not have equal probabilities.
There are 36 equally likely ordered outcomes:
\[ 6 \times 6 = 36 \]
For example, a total of 7 can occur in six ways:
\[ (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \]
Therefore:
\[ P(\text{sum is 7}) = \frac{6}{36} = \frac{1}{6} \]
A total of 2 can occur only in one way, \((1,1)\):
\[ P(\text{sum is 2}) = \frac{1}{36} \]
| Target sum | Ways to obtain it | Probability |
|---|---|---|
| 2 or 12 | 1 | \(1/36\) |
| 3 or 11 | 2 | \(2/36\) |
| 4 or 10 | 3 | \(3/36\) |
| 5 or 9 | 4 | \(4/36\) |
| 6 or 8 | 5 | \(5/36\) |
| 7 | 6 | \(6/36\) |
A player stakes ₹100 that the total will be 2.
\[ p = \frac{1}{36} \]
Fair gross payout:
\[ G = \frac{100}{1/36} = \text{₹3,600} \]
The player receives a net profit of ₹3,500 if the total is 2.
\[ G = \frac{100(0.95)}{1/36} = \text{₹3,420} \]
The player earns ₹3,320 net on a successful prediction of total 2, rather than ₹3,500.
from secrets import randbelow
STAKE = 100
CHOSEN_FACE = 4
win_probability = 1 / 6
gross_payout = STAKE / win_probability # ₹600 in a fair game
roll = randbelow(6) + 1
if roll == CHOSEN_FACE:
player_net = gross_payout - STAKE
print(f"Rolled: {roll}")
print(f"You won ₹{player_net:.2f}")
else:
player_net = -STAKE
print(f"Rolled: {roll}")
print(f"You lost ₹{-player_net:.2f}")Rolled: 2
You lost ₹100.00The program uses secrets.randbelow(6) + 1 to generate a uniformly distributed die result from 1 to 6.
This version uses the same unbiased die roll, but sets a 5% house edge by reducing the payout.
from secrets import randbelow
STAKE = 100
HOUSE_EDGE = 0.05
CHOSEN_FACE = 4
win_probability = 1 / 6
gross_payout = STAKE * (1 - HOUSE_EDGE) / win_probability
net_win = gross_payout - STAKE
roll = randbelow(6) + 1
if roll == CHOSEN_FACE:
print(f"Rolled: {roll}")
print(f"You won ₹{net_win:.2f}")
else:
print(f"Rolled: {roll}")
print(f"You lost ₹{STAKE:.2f}")Rolled: 5
You lost ₹100.00The die remains unbiased. The house edge comes from the stated payout—₹570 rather than the fair ₹600—not from secretly changing the result.
from secrets import randbelow
NUM_ROLLS = 10_000
STAKE = 100
HOUSE_EDGE = 0.05
CHOSEN_FACE = 4
win_probability = 1 / 6
gross_payout = STAKE * (1 - HOUSE_EDGE) / win_probability
net_win = gross_payout - STAKE
player_profit = 0.0
casino_profit = 0.0
wins = 0
for _ in range(NUM_ROLLS):
roll = randbelow(6) + 1
if roll == CHOSEN_FACE:
wins += 1
player_profit += net_win
casino_profit -= net_win
else:
player_profit -= STAKE
casino_profit += STAKE
total_wagered = NUM_ROLLS * STAKE
observed_win_rate = wins / NUM_ROLLS
observed_house_edge = casino_profit / total_wagered
theoretical_casino_profit = total_wagered * HOUSE_EDGE
print("DICE CASINO SIMULATION")
print("-" * 35)
print(f"Number of rolls: {NUM_ROLLS:,}")
print(f"Stake per roll: ₹{STAKE:.2f}")
print(f"House edge: {HOUSE_EDGE:.2%}")
print(f"Gross payout on a win: ₹{gross_payout:.2f}")
print(f"Number of player wins: {wins:,}")
print(f"Observed win rate: {observed_win_rate:.2%}")
print()
print(f"Total amount wagered: ₹{total_wagered:,.2f}")
print(f"Player net result: ₹{player_profit:,.2f}")
print(f"Casino profit: ₹{casino_profit:,.2f}")
print(f"Observed house edge: {observed_house_edge:.2%}")
print(f"Expected casino profit: ₹{theoretical_casino_profit:,.2f}")DICE CASINO SIMULATION
-----------------------------------
Number of rolls: 10,000
Stake per roll: ₹100.00
House edge: 5.00%
Gross payout on a win: ₹570.00
Number of player wins: 1,670
Observed win rate: 16.70%
Total amount wagered: ₹1,000,000.00
Player net result: ₹-48,100.00
Casino profit: ₹48,100.00
Observed house edge: 4.81%
Expected casino profit: ₹50,000.00The theoretical casino profit is:
\[ 10,000 \times \text{₹100} \times 0.05 = \text{₹50,000} \]
The simulation will not necessarily produce exactly ₹50,000. Random variation means that 10,000 rolls may yield a higher or lower casino profit. Repeating the experiment, or increasing the number of rolls, should make the observed edge tend closer to 5%.
A fair chance game has an expected player gain of zero; its payout exactly matches the probability of a win. A casino-style game can still be procedurally fair—random outcomes, visible rules, and consistent payouts—while giving the operator an advantage through a lower-than-fair payout.
The key design principle is:
\[ \text{Fair payout} = \frac{\text{Stake}}{\text{Probability of winning}} \]
\[ \text{Casino payout} = \frac{\text{Stake} \times (1 - \text{House edge})}{\text{Probability of winning}} \]
In a transparent simulation, randomness determines each outcome, while the published payout schedule determines whether the game is mathematically fair or has a house edge.