A sum of such bets, whose probabilities are modelled with a Uniform or Beta distribution, becomes Gaussian rapidly.

This is a mathematical claim: a sum of independent binary bets, each paying off with probability p, converges rapidly to a Gaussian distribution. True: standard results (the central limit theorem for bounded variables, Berry–Esseen bounds) show that such sums approach normality quickly, so that the approximation is good after a modest number of bets for typical values of p. Mostly true: convergence is rapid except when p is very close to 0 or 1, where it is noticeably slower, and the claim omits that caveat. Mixed: convergence is rapid only under restrictive conditions that the claim does not state. Mostly false: convergence occurs but is generally slow. False: such sums do not converge to a Gaussian.

Set before the research began.

1. E1 Berry–Esseen theorem - Wikipedia Wikipedia · 6 March 2004

   “In the case of independent samples, the convergence rate is n−1/2, where n is the sample size, and the constant is estimated in terms of the third absolute normalized moment.”

2. E2 Berry–Esseen theorem - Wikipedia Wikipedia · 6 March 2004

   “Note that the approximation error for all n (and hence the limiting rate of convergence for indefinite n sufficiently large) is bounded by the order of n−1/2.”

3. E3 Error in the normal approximation to the binomial distribution John D. Cook · 17 October 2026

   “However, the function (p2 + q2) /√(pq) is unbounded as p approaches either 0 or 1.”

4. E4 Error in the normal approximation to the binomial distribution John D. Cook · 17 October 2026

   “For a binomial(10, 0.1) random variable, the maximum error in the normal approximation is 0.05 even when using the continuity correction.”

E1 and E2 say that the Berry–Esseen theorem bounds the error of the normal approximation for sums of independent variables at order n^(-1/2). That supports the claim that sums of bounded binary bets become Gaussian quickly. However, E3 shows that when this bound is applied to binomial sums, it grows without limit as p approaches 0 or 1. E4 gives a worked example: with n=10 and p=0.1, the maximum error is still 0.05. So convergence is fast for typical p but noticeably slower near the extremes, a caveat the claim leaves out, which fits the 'mostly true' criterion.