rules: 37 numbers each numbers pay 36 times. play x unit and everytime a number hit that number start over to phase one. phase one: play 1 unit 30 times. phase two: play 3 unit 20 times. phase three: play 5 units 15 times. phase four: play 10 units 15 times. phase five: play 15 units 15 times. phase six: play 30 units 15 times
take in consideration that when i play a number 1 unit 30 times on phase one and play 3 units on phase two and hit on phase two i make profit on that number. phase 1: 30 x 1 unit= 30 units. phase two: 20 x 3 units= 60 units. total lost 90 units. number hit on the 20th round of phase two is 3 units x 36= 108. 108 - 90= 18 unit profits.

You're describing a betting strategy for a roulette game with 37 numbers. In this strategy, you progressively increase your bets through six phases, and you start over whenever your selected number hits. Let's break down the strategy mathematically and verify the profit or loss at each phase:

### Strategy Breakdown:
1. **Phase 1:**
   - Bet 1 unit for 30 rounds.
   - Total units bet: \( 30 \times 1 = 30 \) units.
  
2. **Phase 2:**
   - Bet 3 units for 20 rounds.
   - Total units bet: \( 20 \times 3 = 60 \) units.
   - Total cumulative units bet: \( 30 + 60 = 90 \) units.

3. **Phase 3:**
   - Bet 5 units for 15 rounds.
   - Total units bet: \( 15 \times 5 = 75 \) units.
   - Total cumulative units bet: \( 90 + 75 = 165 \) units.

4. **Phase 4:**
   - Bet 10 units for 15 rounds.
   - Total units bet: \( 15 \times 10 = 150 \) units.
   - Total cumulative units bet: \( 165 + 150 = 315 \) units.

5. **Phase 5:**
   - Bet 15 units for 15 rounds.
   - Total units bet: \( 15 \times 15 = 225 \) units.
   - Total cumulative units bet: \( 315 + 225 = 540 \) units.

6. **Phase 6:**
   - Bet 30 units for 15 rounds.
   - Total units bet: \( 15 \times 30 = 450 \) units.
   - Total cumulative units bet: \( 540 + 450 = 990 \) units.

### Profit Calculation for Each Phase:

1. **Phase 1:**
   - If you hit on the last round of Phase 1 (30th round), the payout is \( 1 \times 36 = 36 \) units.
   - Profit: \( 36 - 30 = 6 \) units.

2. **Phase 2:**
   - If you hit on the last round of Phase 2 (20th round), the payout is \( 3 \times 36 = 108 \) units.
   - Profit: \( 108 - 90 = 18 \) units.

3. **Phase 3:**
   - If you hit on the last round of Phase 3 (15th round), the payout is \( 5 \times 36 = 180 \) units.
   - Profit: \( 180 - 165 = 15 \) units.

4. **Phase 4:**
   - If you hit on the last round of Phase 4 (15th round), the payout is \( 10 \times 36 = 360 \) units.
   - Profit: \( 360 - 315 = 45 \) units.

5. **Phase 5:**
   - If you hit on the last round of Phase 5 (15th round), the payout is \( 15 \times 36 = 540 \) units.
   - Profit: \( 540 - 540 = 0 \) units (no profit, break-even).

6. **Phase 6:**
   - If you hit on the last round of Phase 6 (15th round), the payout is \( 30 \times 36 = 1080 \) units.
   - Profit: \( 1080 - 990 = 90 \) units.

### Summary of Strategy:
- **Phase 1:** \( +6 \) units
- **Phase 2:** \( +18 \) units
- **Phase 3:** \( +15 \) units
- **Phase 4:** \( +45 \) units
- **Phase 5:** \( 0 \) units (break-even)
- **Phase 6:** \( +90 \) units

The strategy indeed results in profit if you hit during any of the phases, with the profit generally increasing as you progress through the phases. The break-even point occurs in Phase 5, after which hitting will result in significant profit.

Do you have any further questions or want details on any part of the analysis? 

### Related Questions:
1. What is the probability of hitting a specific number in a single round of roulette?
2. How does this strategy compare to other betting strategies in roulette?
3. What is the risk of losing all bets before hitting a number?
4. Can this strategy be adjusted to increase profits in earlier phases?
5. What are the long-term expectations for using this strategy in a casino?
6. How does the house edge affect the expected outcomes of this strategy?
7. What are the odds of progressing to each phase without hitting a number?
8. How would adding more phases impact the overall risk and reward?

### Tip:
Always consider the house edge when planning betting strategies; over the long term, the casino's advantage can erode profits from any betting system.

Explore a detailed analysis of a progressive betting strategy in roulette, analyzing six phases with increasing bet sizes. Understand the probability calculations and expected profits for each phase. Suitable for advanced learners and casino strategy enthusiasts.

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