Summary: Consider a probability measure \(\mu\) on [0,1] and an independent sequence of random variables \(X_ 1,\cdot \cdot \cdot,X_ n,\cdot \cdot \cdot\) distributed according to \(\mu\). No regularity assumptions are made on \(\mu\). Denote by \(F_ n(X_ 1,\cdot \cdot \cdot,X_ n)\) the number of unit-size bins that are used by First Fit Decreasing to pack \(X_ 1,\cdot \cdot \cdot,X_ n\). The existence of a constant f(\(\mu)\) is proven such that for each \(\epsilon >0\), we have \[ \sum_{n\geq 1}P(| n^{-1}F_ n(X_ 1,\cdot \cdot \cdot,X_ n)-f(\mu)| \geq \epsilon)<\infty. \]