probability

Probability to win at least once

Our intuition is often wrong about probabilities. Sometimes people think for example that with a probability $\frac{1}{3}$ of winning, if they try $3$ times it is almost sure to win at least once. It is really wrong.

• Let’s the probability to win $\frac{1}{3}\simeq 33.33\mathrm{\%}$ (i.e. $1$ chance in $3$ to win).
• So the probability to lose is $1-\frac{1}{3}\simeq 66.67\mathrm{\%}$.
• The probability to lose each times on $3$ tries is $(1-\frac{1}{3}{)}^3\simeq 29.63\mathrm{\%}$.
• The probability to win at least once on $3$ tries is $1-(1-\frac{1}{3}{)}^3\simeq 70.37\mathrm{\%}$.

More generally for $N$:

• Let’s the probability to win $\frac{1}{N}$ (i.e. $1$ chance in $N$ to win).
• So the probability to lose is $1-\frac{1}{N}$.
• The probability to lose each times on $N$ tries is $(1-\frac{1}{N}{)}^N$.
• The probability to win at least once on $N$ tries is $1-(1-\frac{1}{N}{)}^N$.

The limit when $N$ tends to infinity: $\underset{N\to \mathrm{\infty }}{lim}1-(1-\frac{1}{N}{)}^N=1-\frac{1}{e}\simeq 63.21\mathrm{\%}$.

Links

• The JavaScript file used by this application: probability.js
• This application uses these other free softwares:
  • Chart.js
  • MathJax
• Online tool to compute the probability of S successes on N draws