 Birthday collision count

The birthday collision count counts equal-label unordered pairs. Write it as a sum of indicator random variables:This uses linearity of expectation; mutual independence of the pair indicators is unnecessary. For 23 uniform calendar labels, $\mathbb{E}[C]=253/365\approx 0.69315$, whereas $\mathbb{P}(C\ge 1)\approx 0.50730$. A configuration with three equal labels contributes three pairs, so counting pairs and recording whether any pair exists are different random variables.

Treating every pair event as mutually independent would give the wrong exact formula. For three draws, $X_1=X_2$ and $X_1=X_3$ force $X_2=X_3$. The probability that all three equalities hold is $1/m^2$, while the product of their individual probabilities is $1/m^3$ for $m>1$. This distinguishes independence of the original label draws from mutual independence of events built from overlapping draws.