To determine how many item boxes you need to open to have a certain probability of getting the best item with a 0.2% chance (or 0.002 in decimal form), you can use the concept of expected probability.
If you want to know how many boxes you need to open to have at least a certain probability of getting the item, you can use the formula for the complement probability:
- Let \( p \) be the probability of success (getting the item): \( p = 0.002 \).
- Let \( n \) be the number of attempts (item boxes opened).
- The probability of not getting the item in one attempt is \( 1 - p = 0.998 \).
- The probability of not getting the item in \( n \) attempts is \( (1 - p)^n = (0.998)^n \).
- Therefore, the probability of getting at least one item in \( n \) attempts is \( 1 - (0.998)^n \).
To find \( n \) for a certain probability threshold (let's say you want at least a 50% chance of getting the item), you can set up the inequality:
\[ 1 - (0.998)^n \geq 0.5 \]
Solving this:
\[ (0.998)^n \leq 0.5 \]
Taking the natural logarithm of both sides:
\[ n \cdot \ln(0.998) \leq \ln(0.5) \]
Now, calculate \( n \):
\[ n \geq \frac{\ln(0.5)}{\ln(0.998)} \]
Calculating the values:
- \( \ln(0.5) \approx -0.693147 \)
- \( \ln(0.998) \approx -0.002002 \)
Now plug in these values:
\[ n \geq \frac{-0.693147}{-0.002002} \approx 346.257 \]
Thus, you would need to open approximately 347 item boxes to have at least a 50% chance of getting the item with a 0.2% drop rate.
You can adjust the percentage of success you're aiming for by repeating this calculation with a different threshold.