Probability

Core Theoretical Principles

Probability theory formalizes uncertainty and random processes. Our intuition regularly misjudges odds due to cognitive biases.

1. The Birthday Paradox

In a group of $n$ people, what is the probability $P(n)$ that at least two people share a birthday? It is easier to calculate the complementary probability $P'(n)$ that all birthdays are distinct:

$$ P'(n) = 1 \times \left(1 - \frac{1}{365}\right) \times \left(1 - \frac{2}{365}\right) \times \dots \times \left(1 - \frac{n-1}{365}\right) = \prod_{k=0}^{n-1} \left(1 - \frac{k}{365}\right) $$

Using the Taylor approximation $1 - x \approx e^{-x}$ for small $x$:

$$ P'(n) \approx \prod_{k=0}^{n-1} e^{-k/365} = \exp\left(-\frac{1}{365} \sum_{k=0}^{n-1} k\right) = \exp\left(-\frac{n(n-1)}{2 \times 365}\right) $$

Thus:

$$ P(n) = 1 - P'(n) \approx 1 - e^{-\frac{n(n-1)}{730}} $$

For $n = 23$ individuals, the number of distinct pairs is $\binom{23}{2} = 253$, resulting in:

$$ P(23) \approx 1 - e^{-253/730} \approx 50.7\% $$

Counter-intuitively, only 23 people are required to reach an even chance of a shared birthday.