## 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 33.3\%@@ (i.e. @@1@@ chance in @@3@@ to win).
- So the probability to lose is @@1 - \frac{1}{3} \simeq 66.7\%@@.
- The probability to lose each times on @@3@@ tries is @@(1 - \frac{1}{3})^3 \simeq 29.6\%@@.
- The probability to win at least once on @@3@@ tries is @@1 - (1 - \frac{1}{3})^3 \simeq 70.4\%@@.

More generally:

- 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@@.