Probability Simulator / Calculator | Loot box, Lottery

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

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

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

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

$1-0.99^{100}=1-0.36603...=0.63397...\approx63.4\%$
-> 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.

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

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

Probability of not hitting even once:
$P_0={}_{100}C_0 \times 0.01^0 \times (1-0.01)^{(100-0)}\\ =1 \times 1 \times 0.99^{100}\\ \approx 0.36603$

Probability of hitting only once:
$P_1={}_{100}C_1 \times 0.01^1 \times (1-0.01)^{(100-1)}\\ =100 \times 0.01 \times 0.99^{99}\\ \approx 0.36973$

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