HIROTA YANO
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# Probability Calculator & Simulator / Tool

This is a calculator for "the probability of hitting m times or more after n trials of probability p" and a simulator that "keeps trying until it hits m times" with the set probability.
Settings & Calculate
Probability
(0.01–100%)
Trial
(1–10000)
times
Hiting
(1–100)
times
Probability of hitting 1 times or more
after 100 trials of probability 1%
63.4%
Simulation
A simulation of "keeps trying until probability of 1% hits 1 times" is performed on 100 people.
ChallengerResult
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Within 100
0 people
Avg
0 times
Min
0 times
Max
0 times

## How to use this tool

• Enter the “Probability (percentage or fraction)”, “Trial”, and “Hiting”, and the calculation results will be displayed automatically.
• Probability calculation results are rounded to the third decimal place.

## Calculation of the probability of hitting probability p more than once in n tries

Probability of hitting more than once = 100% - Probability of not hitting once

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

#### e.g. Probability 1%, 1 try, probability of hitting at least 1.

$1-0.99=0.01=1\%$
-> About 1 in 100 hit.

#### e.g. Probability 1%, 100 try, probability of hitting at least 1.

$1-0.99^{100}=1-0.36603...=0.63397...\approx63.4\%$
-> About 2 in 3 hit.

## Calculation of the probability of hitting a probability p more than m times in n tries

If the number of times you want to hit is more than two, the calculation becomes more difficult.

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

The “probability of hitting only x times” can be calculated using the following formula.

$P=nCx \times p^x \times (1-p)^{(n-x)}$

nCx is the number of combinations to choose x out of n different ones.

$nCx={\dfrac {n!}{x!(n-x)!}}$

#### e.g. Probability 1%, 1 try, probability of hitting 2 or more times

Probability of not hitting 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 two or more times:
$P=1-P_0-P_1$
$=1-0.36603-0.36973$
$=0.26424$
$\approx 26.42\%$