Speaker
Description
Tossing a fair coin $N$ times, a gambler wins/loses $10\%$ of his/her holdings against the house if each toss is head/tail ($H$/$T$). Measuring his/her fortunes by $R$ (ratio of final to initial wealth), then we may ask for $\left\langle R\right\rangle $ (the average over all possible $2^{N}$ histories). Since the game sounds like it's even, we may guess $\left\langle R \right\rangle =1$. When computed exactly, it is indeed so. Yet, when simulations are done, $\left\langle R\right\rangle $ drops exponentially with $N$, e.g., to $O\left( 10^{-14}\right) $ for $N=10K$. A further puzzle is the following: It is tempting to apply the central limit theorem and replace the distribution of $H-T$ by a normal (since the exact one is just a binomial). Replying on that leads to an $\left\langle R\right\rangle $ that increases exponentially with $N$! Along with resolutions to these paradoxes, I propose that we add a 'warning label' when the central limit theorem is taught.
