# Empirical Probability Function

• QUOTE: Consider a sample space, $\displaystyle{ S }$, and any event, $\displaystyle{ A }$, defined on S. If our experiment were performed one time, either $\displaystyle{ A }$ or AC would be the outcome. If it were performed $\displaystyle{ n }$ times, the resulting set of sample outcomes would be members of $\displaystyle{ A }$ on $\displaystyle{ m }$ occasions, $\displaystyle{ m }$ being some integer between 0 and $\displaystyle{ n }$, inclusive. Hypothetically, we could continue this process an infinite number of times. As $\displaystyle{ n }$ gets large, the ratio m/n will fluctuate less and less (we will make that statement more precise a little later). The number that m/n convert to is called the empirical probability of $\displaystyle{ A }$ : that is, P(A) = limn→∞(m/n). … the very act of repeating an experiment under identical conditions an infinite number of times is physically impossible. And left unanswered is the question of how large $\displaystyle{ n }$ must be to give a good approximation for limn→∞(m/n) …