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Let us now move on to some basic statistical concepts. How do we know the probabilities for the outcomes of the die roll? In practice, you would observe numerous die rolls and get counts of how many times the various outcomes were observed. Once you have counts, you can divide by the total counts to have the frequency of occurrence of the different outcomes. If you have enough observational data, the frequencies then become better and better estimates of the true underlying probabilities for those outcomes for the system observed (a result due to the law of large numbers (LLN), which is rederived in ssss1). Let us proceed with adding more code in prog1.py that begins with counts on the different die rolls:

------------------ prog1.py addendum 1 ----------------------- rolls = np.array([3435.0,3566,3245,3600,3544,3427]) numterms = len(rolls) total_count = 0 for index in range(0,numterms): total_count += rolls[index] print(total_count) probs = np.array([0.0,0,0,0,0,0]) for index in range(0,numterms): probs[index] = rolls[index]/total_count; print(probs) -------------------- end prog1.py addendum 1 -----------------

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