The celebrated hook length formula (https://en.wikipedia.org/wiki/Hook_length_formula) says that the number of standard Young tableaux of shape $\lambda \vdash n$ is $n! \cdot \prod_{u\in \lambda}h_u^{-1}$ where $h_u$ is the hook length of the box $u$ of $\lambda$.
A heuristic argument (I hesitate to call it a "proof") put forward by Knuth for this formula goes as follows. In a random injective filling of the boxes of $\lambda$ with the numbers $1,2,\ldots,n$, the probability that a given box $u$ has the smallest number in its hook is $h_u^{-1}$. Moreover, such a filling is an SYT if and only if each box is filled with the smallest number in its hook. "Thus", the probability that a random filling is an SYT is $\prod_{u\in \lambda}h_u^{-1}$, and so the number of SYTs is $n! \cdot \prod_{u\in \lambda}h_u^{-1}$. The error with this false proof is of course that the events of boxes being filled with the smallest numbers in their hooks are not independent. But it is quite interesting that we arrive at the correct probability treating these events as independent.