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Independent Drop Odds & Expected Wait

Calculate the probability of at least one success over repeated independent attempts, plus the expected wait from a fixed per-attempt chance.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

With a 50% chance on each independent attempt, three attempts give an 87.5% chance of at least one success. The average wait to the first success is two attempts.

At least one success = 1 − (1 − p)^n; expected attempts to first success = 1 / p

Worked example

Enter these known values and leave the other values blank.

Success chance per independent attempt
50 %
Attempts / unrounded inverse threshold
3
Expected attempts until first success
2
Chance of at least one success
87.5 %

Assumptions and limitations

  • Attempts must be independent and use a constant per-attempt chance. Pity systems, changing odds, guaranteed drops and sampling without replacement require a different model.
  • The finite-wait model requires a chance above 0% and at most 100%, and a positive number of attempts. Expected wait is a long-run average, not a guaranteed deadline.
  • Forward trial counts should be whole numbers for actual attempts. An inverse result can be fractional; round up to a whole count to reach a requested probability and check that count in the forward model.
  • Some boundary targets do not determine a finite or unique inverse. For example, a chance below 100% per attempt does not reach exactly 100% after a finite number of attempts.

Common questions

Does the expected wait guarantee a success by then?

No. For example, a 50% chance has an expected wait of two attempts, but the chance of a success within two attempts is 75%.

Can I use this when the game increases the chance after failures?

No. This formula assumes the chance stays constant and attempts are independent.

References

Bookify requires a positive per-attempt chance for the finite expected-wait model.

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