# Randomized methods

Randomized methods are often used in the probability theory as a kind fo numerical methods to obtain some results which cannot be easily (or even hardly) obtained analytically. One of the most famous examples is Monte Carlo method.

If we talk about problems like "find probability $p$ of an event $A$" then using randomized approach we will get an answer like "the solution is $p^*$ such that $\mathsf{P}\left[|p-p^*|>\delta\right]<\varepsilon$" where $\delta,\varepsilon$ are small. We will call it bounds of the 2nd kind. The bound of the 1st kind are like $|p-p^*|<\gamma$.

So in fact the answer is not strict but gives a probabilistic estimation. This method is for sure useful in statistics when the goal is to find the best approximation (say, you never know the true distribution in the real life).

I just would like to know an opinion if the approach is justified for the pure mathematical problems when the distributions of random processes are given explicitly and one can try to calculate its approximation with bounds of the 1st kind (deterministic) rather than of the 2nd kind (probabilistic)?

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is this a matter of ODE's used to model a situation vs stochastic processes? –  picakhu Apr 7 '11 at 16:06
Sorry, I haven't got your question. –  Ilya Apr 7 '11 at 16:29
en.wikipedia.org/wiki/Mathematical_model Under classifying models, look at point 2. –  picakhu Apr 7 '11 at 16:31