I wonder if the Weak Law of Large Numbers is only applicable if the random variable is binomially distributed. (The random variable counts the relative frequency of an event A). So, when you describe the Law, do you have to mention as a prerequisite that you only look at Bernoulli experiments?
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As long as the $X_i$ are independent identically distributed, and their mean exists, the sample mean converges in probability to the mean. There is no need to have the $X_i$ be Bernoulli.
Check it out here. Both Weak LLN and Strong LLN are stated over any sequence of iid integrable random variables, so it applies not only to Bernoulli random variables, but also to Poisson, Gaussian, exponential etc.