10% per ticket. How many tickets for 90% chance to win?

To achieve a 90% chance of winning in a situation where each ticket has a 10% probability of winning, we can solve this by understanding the probability of *not* winning with each ticket and then finding the number of tickets needed for the cumulative probability to reach 90%.

### Solution:
1. **Define the probability of not winning with each ticket:**  
   Since each ticket has a 10% chance of winning, it has a 90% chance of not winning. This probability of not winning with one ticket is:
   \[
   1 - 0.1 = 0.9
   \]

2. **Calculate the probability of not winning with multiple tickets:**  
   Let \( n \) be the number of tickets. The probability of not winning with \( n \) tickets is \( 0.9^n \).

3. **Set up the equation for a 90% chance of winning:**  
   We want the probability of winning to be 90%, which means the probability of not winning should be 10%.
   \[
   0.9^n = 0.1
   \]

4. **Solve for \( n \) using logarithms:**  
   \[
   n \ln(0.9) = \ln(0.1)
   \]
   \[
   n = \frac{\ln(0.1)}{\ln(0.9)}
   \]
   Calculating this, we get:
   \[
   n \approx \frac{-2.3026}{-0.1054} \approx 21.85
   \]

Since we can't buy a fraction of a ticket, we round up to the next whole number: **22 tickets**.

### Answer:
You need to buy **22 tickets** to have at least a 90% chance of winning.

Would you like more details or have any questions about this calculation?

### Follow-up Questions:
1. What if each ticket had a 15% chance of winning? How many tickets for a 90% chance?
2. How many tickets are needed for a 99% chance of winning if each has a 10% win rate?
3. What is the probability of winning if you buy only 10 tickets at a 10% chance each?
4. How does changing the win rate per ticket affect the number needed for a 90% overall chance?
5. How would the answer change if tickets had independent probabilities of winning?

### Tip:
Using logarithms is essential for solving problems involving exponential growth or decay, especially in probability and statistics.

Discover how many tickets are needed to achieve a 90% chance of winning when each ticket has a 10% probability of success. This solution explains key probability concepts, exponential decay, and logarithmic calculations, making it suitable for students in Grades 10-12.

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