Question
if each box costs 300 and a legendary hero mask has a 0.5% chance of coming out the box how many boxes would you have to open, there is also another legendary influencer monkey that can come out that same box at 0.5% chance of dropping as well acount for the cost of boxes
Answers
GPT-4o mini
To calculate how many boxes you would need to open to have a good chance of obtaining at least one legendary hero mask or legendary influencer monkey, we can use the concept of probabilities and expected value.
1. **Probability of Getting at Least One Item**: The probability of not getting either the hero mask or the influencer monkey from one box is:
- Probability of not getting hero mask = \(1 - 0.005 = 0.995\)
- Probability of not getting influencer monkey = \(1 - 0.005 = 0.995\)
- Probability of not getting either = \(0.995 \times 0.995 = 0.990025\)
2. **Finding the Probability of Getting at Least One Item After Opening \(n\) Boxes**:
- The probability of not getting either after \(n\) boxes is \(0.990025^n\).
- Therefore, the probability of getting at least one is \(1 - 0.990025^n\).
3. **Deciding on a Target Probability**: Let's say you want at least a 50% chance of getting at least one of the items. We can set up the equation:
\[
1 - 0.990025^n \geq 0.5
\]
\[
0.990025^n \leq 0.5
\]
4. **Solving for \(n\)**:
\[
n \log(0.990025) \leq \log(0.5)
\]
\[
n \geq \frac{\log(0.5)}{\log(0.990025)}
\]
Using the logarithm base 10 or natural logarithm, you can calculate:
\[
n \geq \frac{-0.3010}{-0.0043} \approx 69.86
\]
This means you would need to open approximately 70 boxes to have at least a 50% chance of receiving either the legendary hero mask or the legendary influencer monkey.
5. **Cost Calculation**: The cost of each box is $300, so the total cost for opening \(n\) boxes would be:
\[
\text{Cost} = n \times 300 = 70 \times 300 = 21000
\]
### Conclusion:
You would need to open approximately **70 boxes**, which would cost you **$21,000** to have at least a 50% chance of obtaining either a legendary hero mask or a legendary influencer monkey.
1. **Probability of Getting at Least One Item**: The probability of not getting either the hero mask or the influencer monkey from one box is:
- Probability of not getting hero mask = \(1 - 0.005 = 0.995\)
- Probability of not getting influencer monkey = \(1 - 0.005 = 0.995\)
- Probability of not getting either = \(0.995 \times 0.995 = 0.990025\)
2. **Finding the Probability of Getting at Least One Item After Opening \(n\) Boxes**:
- The probability of not getting either after \(n\) boxes is \(0.990025^n\).
- Therefore, the probability of getting at least one is \(1 - 0.990025^n\).
3. **Deciding on a Target Probability**: Let's say you want at least a 50% chance of getting at least one of the items. We can set up the equation:
\[
1 - 0.990025^n \geq 0.5
\]
\[
0.990025^n \leq 0.5
\]
4. **Solving for \(n\)**:
\[
n \log(0.990025) \leq \log(0.5)
\]
\[
n \geq \frac{\log(0.5)}{\log(0.990025)}
\]
Using the logarithm base 10 or natural logarithm, you can calculate:
\[
n \geq \frac{-0.3010}{-0.0043} \approx 69.86
\]
This means you would need to open approximately 70 boxes to have at least a 50% chance of receiving either the legendary hero mask or the legendary influencer monkey.
5. **Cost Calculation**: The cost of each box is $300, so the total cost for opening \(n\) boxes would be:
\[
\text{Cost} = n \times 300 = 70 \times 300 = 21000
\]
### Conclusion:
You would need to open approximately **70 boxes**, which would cost you **$21,000** to have at least a 50% chance of obtaining either a legendary hero mask or a legendary influencer monkey.