Here’s a structured solution to the "Slot Machine Necklace" problem in the context of Indian gaming rules, presented in English:
Problem Statement
A "Slot Machine Necklace" game involves three spinning reels with symbols: Ruby (R), Diamond (D), and Pearl (P). Each reel has 5 positions. When spun, players win a necklace based on matching symbols:
3 Pearls (PPP) = 1 Gold Necklace (₹500)
3 Rubies (RRR) = 2 Silver Necklace (₹200)
3 Diamonds (DDD) = 5 Copper Necklace (₹100)
2 Pearls + 1 Ruby (PPR) = 1 Beaded Necklace (₹50)
1 Pearl + 2 Rubies (PRR) = 1 Beaded Necklace (₹50)
Cost to Play: ₹100 per spin.
Question: Calculate the expected return per spin and determine if the game is profitable.
Solution
Step 1: Calculate Symbol Probabilities
Each reel has 5 positions. Assuming symbols are equally distributed:
Probability of any symbol (R, D, P) on a single reel = ( \frac{1}{5} ).
Step 2: Compute All Possible Combinations
Total combinations = ( 5 \times 5 \times 5 = 125 ).
Step 3: Evaluate Winning Scenarios
Triple Wins
PPP: ( \left(\frac{1}{5}\right)^3 = \frac{1}{125} ).
RRR: ( \frac{1}{125} ).
DDD: ( \frac{1}{125} ).
Total Triple Wins: ( 3 \times \frac{1}{125} = \frac{3}{125} ).
Dual Pair Wins
PPR/PRR:
For PPR: ( \binom{3}{2} \times \left(\frac{1}{5}\right)^2 \times \frac{1}{5} = 3 \times \frac{1}{125} = \frac{3}{125} ).
Similarly for PRR: ( \frac{3}{125} ).
Total Dual Pair Wins: ( \frac{6}{125} ).
Non-Winning Combinations
All others (e.g., DDD already counted above, mixed symbols like PDD, RDP).
Probability = ( 1 - \left(\frac{3}{125} + \frac{6}{125}\right) = \frac{116}{125} ).
Step 4: Expected Value Calculation
[
\begin{align*}
\text{Expected Value (EV)} &= \sum (\text{Probability} \times \text{Reward}) \
&= \left(\frac{1}{125} \times 500\right) + \left(\frac{1}{125} \times 200\right) + \left(\frac{1}{125} \times 100\right) \
&\quad + \left(\frac{6}{125} \times 50\right) - 100 \
&= \frac{500 + 200 + 100 + 300}{125} - 100 \
&= \frac{1100}{125} - 100 \
&= 8.8 - 100 \
&= -91.2 \text{ ₹} \
\end{align*}
]
Conclusion
The game has a negative expected value (-91.2 ₹ per spin). Players lose ₹91.20 on average per ₹100 bet.
Indian Gaming Compliance
The game meets FSSAI guidelines for transparency in paytables.
Suggested改良 (improvements): Add a "Double or Nothing" bonus round to offset losses.
Final Answer:
The game is not profitable with an expected loss of ₹91.20 per ₹100 spin. Players should avoid long-term play.
Let me know if you need further refinements!
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