Let’s say there a game with 20 sets of two doors, one door lets you continue and raise you price, at the risk of losing it all, whilst the other one stops you and gives you your reward immediately. So the risk of losing it all also increases by every door it starts at 5% and increases by 5% until it reaches the cap of 50% what would the chance of getting through all the doors be?

The problem describes a scenario where a player faces 20 sets of two doors. In each set, one door allows the player to continue, while the other door ends the game, giving the player a reward. The risk of losing (and not progressing) starts at 5% for the first door and increases by 5% for each subsequent set, up to a cap of 50%.

### Step-by-step breakdown:

1. **Risk of losing** per set:
   - For the first set of doors, there's a 5% risk of losing (95% chance of continuing).
   - For the second set, the risk increases by 5%, making it a 10% chance of losing (90% chance of continuing).
   - This risk continues to increase by 5% for each subsequent set until it reaches a cap of 50%.

2. **Finding the probability of surviving all 20 doors**:
   - The probability of passing through each set is the complement of the risk of losing (i.e., \( P(\text{pass}) = 1 - P(\text{lose}) \)).
   - The overall probability of passing through all 20 doors is the product of the individual probabilities of passing through each set.

Let's compute the total probability of passing all 20 doors:

\[
P(\text{total}) = P_1 \times P_2 \times P_3 \times \cdots \times P_{20}
\]

Where:
- \( P_1 = 95\% = 0.95 \)
- \( P_2 = 90\% = 0.90 \)
- \( P_3 = 85\% = 0.85 \)
- …
- For \( P_1 \) to \( P_{10} \), the risk increases by 5% each time, until it caps at 50% for \( P_{11} \) to \( P_{20} \).

I'll now calculate the total probability.The probability of successfully passing through all 20 doors is approximately \( 0.00003197 \), or about 0.0032%. This indicates a very low chance of making it through all the doors without losing.

Would you like further details or clarification on any part of the solution? Here are some related questions you might find interesting:

1. What would the probability be if the risk increased by 3% instead of 5%?
2. How would the chances change if the number of doors increased to 30?
3. What would happen if the risk capped at 40% instead of 50%?
4. How does the probability change if we start with a 2% risk increment?
5. What if we included a third door that gave a different reward?

**Tip:** When working with compounded probabilities, each individual risk has a significant impact on the overall outcome.

Let’s say there a game with 20 sets of two doors, one door lets you continue and raise your prize, at the risk of losing it all, whilst the other one stops you and gives you your reward immediately. So the risk of losing it all also increases by every door; it starts at 5% and increases by 5% until it reaches the cap of 50%. What would the chance of getting through all the doors be?

Discover how to calculate the probability of passing through 20 doors in a game where the risk of losing starts at 5% and increases by 5% until it caps at 50%. Learn how to use compounded probability to find the total chance of success. Suitable for students in Grades 10-12.

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