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\%@@ (i.e. @@1@@ chance in @@3@@ to win).
- So the probability to lose is @@1 - \frac{1}{3} \simeq 66.67\%@@.
- The probability to lose each times on @@3@@ tries is @@(1 - \frac{1}{3})^3 \simeq 29.63\%@@.
- The probability to win at least once on @@3@@ tries is @@1 - (1 - \frac{1}{3})^3 \simeq 70.37\%@@.
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: @@\lim\limits_{N\rightarrow\infty} 1 - (1 - \frac{1}{N})^N = 1 - \frac{1}{e} \simeq 63.21\%@@.