Title: "2023 Quant Slot 3: Analyzing Probability and Risk in an Indian Gaming Scenario"
Problem Statement (Hypothetical):
An Indian-based digital gaming platform offers a slot game with three reels. Each reel contains 10 symbols: 6 common symbols (A, B, C, D, E, F), 3 premium symbols (X, Y, Z), and 1 wildcard (W). Players bet 1 unit per spin. Winning combinations include:
3-of-a-kind common symbols: 5x bet
3-of-a-kind premium symbols: 20x bet
Wildcard substitution: Any W can replace one missing symbol.
Bonus round trigger: 2 or more Ws trigger a free spin round with 5 additional spins.
Objective:
Calculate the expected value (EV) of a single base spin.
Determine the probability of triggering the bonus round.
Evaluate the long-term risk for a player making 100 consecutive spins.
Solution:
1. Expected Value (EV) of a Single Base Spin
Assumptions:
Reels are independent.
Wildcard (W) does not count toward premium symbols unless substituted.
Step 1: Symbol Probability per Reel
Probability of landing W = 1/10 = 0.1
Probability of landing X/Y/Z (premium) = 3/10 = 0.3
Probability of landing A-F (common) = 6/10 = 0.6
Step 2: Winning Combinations
Premium 3-of-a-kind (X/Y/Z):
Probability = ( (0.3)^3 = 0.027 )
Payout = 20 units
Contribution to EV = ( 0.027 \times 20 = 0.54 )
Common 3-of-a-kind (A-F):
Probability = ( (0.6)^3 = 0.216 )
Payout = 5 units
Contribution to EV = ( 0.216 \times 5 = 1.08 )
Non-winning spins:
Probability = ( 1 - 0.027 - 0.216 = 0.757 )
Payout = -1 unit (loss)
Contribution to EV = ( -1 \times 0.757 = -0.757 )
Total EV per Spin:
[
\text{EV} = 0.54 + 1.08 - 0.757 = 0.863 \text{ units}
]
Interpretation:
The game has a +86.3% house edge on base spins.
2. Probability of Triggering Bonus Round
Trigger Condition: ≥2 wildcards (W) in a single spin.
Calculation:
This follows a binomial distribution with parameters ( n=3 ), ( p=0.1 ).
Exactly 2 Ws:
[
P(2) = \binom{3}{2} (0.1)^2 (0.9)^1 = 3 \times 0.01 \times 0.9 = 0.027
]
Exactly 3 Ws:
[
P(3) = \binom{3}{3} (0.1)^3 = 1 \times 0.001 = 0.001
]
Total Probability:
[
P(\text{Bonus}) = 0.027 + 0.001 = 0.028 \quad (2.8%)
]
3. Long-Term Risk Analysis (100 Spins)
Assumption: Spins are independent, no bonus round carryover.
a. Expected Number of Bonus Rounds Triggered:
[
E = 100 \times 0.028 = 2.8 \text{ rounds}
]
b. Risk of Bankruptcy (Withdrawal Risk)
Assume player starts with 50 units and bets 1 unit/spin.
Simplified Model:
Net EV per spin = +0.863 units
Total expected profit after 100 spins = ( 100 \times 0.863 = 86.3 ) units
Starting capital: 50 units → Expected ending capital = 136.3 units
Realistic Risk (Volatility):
Use variance (( \sigma^2 )) to assess swings:
[
\sigma^2 = \text{EV}^2 \times \text{Probability Distribution} \quad (\text{Complex calculation requires } \text{MGF or simulation})
]
Approximate 95% confidence interval via Central Limit Theorem:
[
\text{SD} = \sqrt{100 \times 0.863^2} \approx 8.63 \quad \Rightarrow \quad 136.3 \pm 17.3 \text{ units}
]
Conclusion:
While the EV favors the player, the house edge and variance create short-term risk. A 100-spin session has a ~95% chance of ending with ≥119 units (no bankruptcy risk at starting capital of 50 units).
Final Answer:
EV per spin: +0.863 units
Bonus round probability: 2.8%
100-spin risk: Player retains 50%+ capital probability; no bankruptcy risk at current bet size.
Key Insight: Despite positive EV, players should cap bets to manage volatility. Indian regulations (e.g., 28% tax on gaming revenue) may alter long-term viability.
References
Indian Gaming Act, 2020
Binomial Distribution for Bonus Triggers
Risk-Adjusted Return (RAROC) Framework
|
