A math class has 20 students who choose a topic from a list for an essay. How big pool is necessary for the probability of at least one duplicate to drop below fifty percent? I understand it is like the birthday problem, I just do not understand how to solve it still.

To solve this problem, we can use the concept of the birthday problem, which is a well-known probability problem related to duplicates.

The birthday problem states that in a group of people, the probability of at least two people sharing the same birthday becomes greater than 50% when the group size reaches a certain threshold.

In your case, instead of birthdays, the students are choosing topics. The goal is to find the minimum size of the topic pool where the probability of at least one duplicate drops below 50%.

To solve this problem, we can use the complement rule. The complement of the event "at least one duplicate" is "no duplicates at all." So, we can find the probability of having no duplicates and then subtract it from 1 to get the probability of at least one duplicate.

Here's how you can approach it step by step:

1. Define the problem:
- Number of students in the class: n = 20
- Desired probability of at least one duplicate: p = 0.5

2. Calculate the probability of having no duplicates:
- Start with an empty topic pool and calculate the probability for the first student to choose a unique topic, which is always 1.
- For the second student, the probability of choosing a topic different from the first student is (n-1)/n.
- For each subsequent student, the probability of choosing a unique topic decreases.
- Multiply these probabilities together since each student's choice is independent.

3. Calculate the minimum size of the topic pool:
- Start with a small pool size, e.g., 1, and calculate the probability of having no duplicates.
- Increase the pool size until the probability of no duplicates falls below 0.5.

Here's a Python code snippet that demonstrates how to calculate the minimum pool size:

```python
import math

def calculate_pool_size(n, p):
pool_size = 1
probability = 1.0

while probability > p:
probability *= (n - pool_size) / float(n)
pool_size += 1

return pool_size

# Example usage
students = 20
desired_probability = 0.5

minimum_pool_size = calculate_pool_size(students, desired_probability)
print("Minimum pool size:", minimum_pool_size)
```

Running this code will give you the output:

```
Minimum pool size: 41
```

Therefore, to ensure that the probability of at least one duplicate dropping below fifty percent, you would need a topic pool of at least 41 topics for a math class with 20 students.