To determine how many times you need to roll to have a good chance of obtaining an item with a 0.05% chance (or 0.0005 as a decimal), you can use the concept of probability.
First, let's define what you mean by a "good chance." A common threshold used is a 50% chance, but you can adjust this depending on your preferences.
Formula
The probability \( P \) of not getting the item in one roll is: \[ P_{\text{not getting}} = 1 - P_{\text{getting}} \] \[ P_{\text{not getting}} = 1 - 0.0005 = 0.9995 \]
If you roll \( n \) times, the probability of not getting the item in any of those rolls is: \[ (P_{\text{not getting}})^n = (0.9995)^n \]
To find out how many rolls would give you at least a 50% chance of getting at least one item, you set up the equation: \[ 1 - (0.9995)^n \geq 0.5 \]
Solving the Inequality
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Rearranging the inequality gives: \[ (0.9995)^n \leq 0.5 \]
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Taking the natural logarithm of both sides: \[ n \cdot \ln(0.9995) \leq \ln(0.5) \]
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Solving for \( n \): \[ n \geq \frac{\ln(0.5)}{\ln(0.9995)} \]
Calculation:
Using a calculator, we find:
- \( \ln(0.5) \approx -0.693147 \)
- \( \ln(0.9995) \approx -0.000500 \)
Now, \[ n \geq \frac{-0.693147}{-0.000500} \approx 1386.294 \]
Conclusion:
You would need to roll approximately 1387 times to have at least a 50% chance of obtaining the item with a 0.05% chance on each roll. If you want a different probability threshold (e.g., 75% or 90%), you can use the same formula and adjust accordingly.