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PART III REVIEW

Some phenomena are random. That is, although their individual outcomes are unpredictable, there is a regular pattern in the long run. Gambling devices (rolling dice, spinning roulette wheels) and taking an SRS are examples of random phenomena. Probability and expected value give us a language to describe randomness. Random phenomena are not haphazard or chaotic any more than random sampling is haphazard. Randomness is instead a kind of order in the world, a long-run regularity as opposed to either chaos or a determinism that fixes events in advance. Chapter 17 discusses the idea of randomness, Chapter 18 presents basic facts about probability, and Chapter 20 discusses expected values.

When randomness is present, probability answers the question, “How often in the long run?’’ and expected value answers the question, “How much on the average in the long run?’’ The two answers are connected because the definition of “expected value’’ is in terms of probabilities. Much work with probability starts with a probability model that assigns probabilities to the basic outcomes. Any such model must obey the rules of probability. Another kind of probability model uses a density curve such as a Normal curve to assign probabilities as areas under the curve. Personal probabilities express an individual’s judgment of how likely some event is. Personal probabilities for several possible outcomes must also follow the rules of probability if they are to be consistent with each other.

To calculate the probability of a complicated event without using complicated math, we can use random digits to simulate many repetitions. You can also find expected values by simulation. Chapter 19 shows how to do simulations. First give a probability model for the outcomes, then assign random digits to imitate the assignment of probabilities. The table of random digits now imitates repetitions. Keep track of the proportion of repetitions on which an event occurs to estimate its probability. Keep track of the mean outcome to estimate an expected value.