Probability Simulator / Calculator | Loot box, Lottery

  • The calculation result of the probability is rounded off to the third decimal place.
日本語 / English

Probability calculation

Probability:
%
/
(0.01% ~ 100%)
Number of trials: times
(1 ~ 1000)
Required hits: times
(1 ~ 100)
Probability of hitting 1 or more times when performed 1% loot box 100 times
63.4%

Simulation

Simulate "Continue to trial until 1% chance hits 1 times" for 100 people.
ChallengerResult
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Min: 0 times
Max: 0 times
Avg: 0 times
Within 100 times: 0 people

Probability of hitting 1 or more times when performed p% loot box n times

Probability of hitting 1 or more = 100% - Probability of not hitting even once

P=1(1p)nP=1-(1-p)^n

e.g. Probability of hitting 1 or more times when performed 1% loot box 1 times

10.99=0.01=1%1-0.99=0.01=1\%
-> It hits about 1 in 100 people.

e.g. Probability of hitting 1 or more times when performed 1% loot box 100 times

10.99100=10.36603...=0.63397...63.4%1-0.99^{100}=1-0.36603...=0.63397...\approx63.4\%
-> It hits about 2 in 3 people.

Probability of hitting m or more times when performed p% loot box n times

If you want to hit more than 2 times, the calculation will be difficult.

  • Probability of hitting more than 2 times = 100% - Probability of not hitting even once - Probability of hitting only once
  • Probability of hitting more than 3 times = 100% - Probability of not hitting even once - Probability of hitting only once - Probability of hitting only twice

“Probability of hitting only x times” can be calculated by the following formula.

P=nCx×px×(1p)(nx)P=nCx \times p^x \times (1-p)^{(n-x)}

nCx is number of combinations to choose x from n different.

nCx=n!x!(nx)!nCx={\dfrac {n!}{x!(n−x)!}}

e.g. Probability of hitting 2 or more times when performed 1% loot box 100 times


Probability of not hitting even once:
P0=100C0×0.010×(10.01)(1000)=1×1×0.991000.36603P_0={}_{100}C_0 \times 0.01^0 \times (1-0.01)^{(100-0)}\\ =1 \times 1 \times 0.99^{100}\\ \approx 0.36603

Probability of hitting only once:
P1=100C1×0.011×(10.01)(1001)=100×0.01×0.99990.36973P_1={}_{100}C_1 \times 0.01^1 \times (1-0.01)^{(100-1)}\\ =100 \times 0.01 \times 0.99^{99}\\ \approx 0.36973

Probability of hitting 2 or more times:
P=1P0P1=10.366030.36973=0.2642426.42%P=1-P_0-P_1\\ =1-0.36603-0.36973\\ =0.26424\\ \approx 26.42\%